Finite Schematizable Algebraic Logic
Finite Schematizable Algebraic Logic Ildiko Sainy, Viktor Gyurisz November 25, 1996 (Version 5.2) Abstract
In this work, we attempt to alleviate three (more or less) equivalent negative results. These are (i) non-axiomatizability (by any nite schema) of the valid formula schemas of rst order logic, (ii) non-axiomatizability (by nite schema) of any propositional logic equivalent with classical rst order logic (i.e., modal logic of quanti cation and substitution), and (iii) non-axiomatizability (by nite schema) of the class of representable cylindric algebras (i.e., of the algebraic counterpart of rst order logic). Here we present two nite schema axiomatizable classes of algebras that contain, as a reduct, the class of representable quasi-polyadic algebras and the class of representable cylindric algebras, respectively. We establish positive results in the direction of nitary algebraization of rst order logic without equality as well as that with equality. Finally, we will indicate how these constructions can be applied to turn negative results (i), (ii) above to positive ones.
Contents
1 Introduction 2 Results 2.1 2.2 2.3 2.4 2.5 2.6
Preliminaries : : : : : : : : : : : : : : : : : : : : : : : Finitizing algebras of rst order logic without equality Finitizing algebras of rst order logic with equality : : Open Qusetions : : : : : : : : : : : : : : : : : : : : : : Application to logic : : : : : : : : : : : : : : : : : : : : Comparisons with the polyadic notion of a schema : :
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This is a pre-publication version of the paper, nal version will be prepared later and published elsewhere. y [email protected] z [email protected] or [email protected]. Research supported by Hungarian Research Fund, grants F17452, T16448.
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3 Proofs 3.1 3.2 3.3 3.4 3.5 3.6 3.7
An axiomatization of IGwdfG : : : : : : : : : : : : Theorem 2.7 (1): nite schema axiomatizing IGwdfG Theorem 2.11: compressing the unit element : : : : Quasi-axiomatizing ICdfH : : : : : : : : : : : : : : : Theorem 2.30: the cylindric reduct of Mod(H 1; H 2) Further observations on Mod(H 1; H 2) : : : : : : : : The witness tree method : : : : : : : : : : : : : : : :
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1 Introduction Since it was introduced, rst order logic became the most frequently used and studied logical system. Its logical connectives and basic concepts are motivated by intuitive \mathematical" reasoning. It also plays a fundamental role in logical (formal) investigations of not necessarily mathematical reasoning, cf. e.g. [GG85], [vB91]. First order logic is only one of the logical systems investigated in modern logic. For many logical systems, or logics for short, the methodology of algebraic logic proved to be very useful. This methodology is summarized e.g. in [ANSK] and in [AN94]. Cf. also Blok{Pigozzi [BP89]. The key step in this methodology is that, to any logic L, we associate a class Alg(L) of algebras, called the algebraic counterpart of the logic L. In many cases, investigating Alg(L) is easier than investigating L, e.g. because, in the study of Alg(L), we can use the well developed tools of algebra. A number of problems concerning certain logics L were often solved by translating the problem to Alg(L), solving the algebraic problem by the methodology of algebra, and then translating the algebraic result back to the logical context in which L was originally investigated. Another equally important value of algebraization is obtaining insights into the essential nature of a logic L, via abstraction provided by algebraization. The algebraic counterpart of classical propositional logic is the class of Boolean algebras. Boolean algebras proved immensely useful in studying classical propositional logic, and in understanding its connections with other logical systems. Similarly, for a great number of other logics, studying their algebraic counterparts has proven very useful. One example is propositional modal logic, the algebraic counterpart of which is Boolean algebras with operators. Similar positive examples are most of the non-classical propositional logics, e.g. intuitionistic propositional calculus whose algebraic counterpart is the variety of Heyting Algebras. Cf. e.g. [W88]. In the case of rst order logic, when trying to apply the methodology of algebraic logic in the above outlined fashion, we have to face the following problem. As we said, the algebraic counterpart of classical propositional logic is the class of Boolean algebras. This class turns out to be a nitely axiomatizable equational class. The same is true for the other successful examples we referred to. In each 2
case, Alg(L) is axiomatizable by a nite set of equations or quasi-equations1. As a contrast, in the case of rst order logic, all the algebraic counterparts found before 1987 are very far from being nitely axiomatizable. Namely, J. D. Monk proved in [M70] that the most thoroughly investigated algebraic counterpart of rst order logic with equality, the class RCA! of representable cylindric algebras is not axiomatizable by a nite schema of equations2. A similar result was obtained in Sain{Thompson [SaT91] about the class RQPA! of representable quasi-polyadic algebras (the algebraic counterpart of rst order logic without equality)3 . Monk's theorem was improved by Andreka in [A91]. She proved that any axiomatization of RCA! , besides being in nite, has to be extremely complex. These results motivate the problem of nding an algebraization of rst order logic where the class of algebras associated to this logic is axiomatizable by a preferably nice (simple, transparent, mathematically elegant) schema of equations or quasi-equations. There is a harder version of this problem asking for a strictly nite axiomatization. However, it is understood that a nite schema solution could already be very useful if the schema of axioms is simple enough. This problem is known as the nitization problem in algebraic logic. Actually, as it turns out, it is not a single problem, but rather a family of problems or, perhaps, even a research direction of algebraic logic, cf. e.g. Nemeti [N93], Simon [Si93], Sain [Sa92]. Here we will simply refer to it as the nitization problem. The problem as we formulated above is not concrete enough to be regarded as a tangible mathematical problem. To illustrate this, we note that Simon [Si93] is devoted to showing that many seemingly relevant results are, though mathematically interesting, not even partial solutions of the nitization problem. Therefore, below we recall a suciently concrete formulation of the nitization problem from the literature. The problem of nding a \ nitizable" algebraization of rst order logic was raised in Monk [M70, p.20] and in Henkin{Monk [HM74].4 The formulation in the latter goes as follows: \Devise an algebraic version of predicate logic in which the class of representable algebras forms a nitely based equational class." An equational class is called nitely based if it is axiomatizable by nitely many equations. In [HM74] an algebra is called representable if it is isomorphic to a 1 Quasi-equations are equational implications. 2 Monk obtained similar negative results for RCA3 already in 1964. 3 The Sain{Thompson result was preceded by a relevant result of James Johnson [Jo69]
who proved that RQPAn (2 < n < !) is not nitely axiomatizable. 4 This problem has a long history and many people worked on it, e.g. A. Tarski did so already in the 1940's; but the explicit formulation we would like to use is in the quoted papers of L. Henkin and J. D. Monk. For completeness, we also refer to Henkin{Tarski (1961) [HT61] to the open problem formulated below Theorem 1.14 therein (lines 21{29 on page 96). There they ask for a nite schema axiomatization of Eq(RCA! ). This turns out possible (but only) if we add to RCA! operations like ssuc ; spred de ned herein.
3
subdirect product of set algebras . Therefore the key step is nding an appropriate notion of set algebra. In devising such de nitions, usually, an ordinal is xed in advance, and then the elements of a set algebra are (certain) -ary relations over some set called the base of the algebra. One piece of the connection with logic is that the base of the algebra corresponds to the universe of that model of rst order logic from which it is obtained (see e.g. [HMTII, sections 4.3, 5.6] for details on this connection). Following Jonsson and Tarski, we require that the operations of a set algebra should be invariant under any permutation of the base set5 . An operation with this property is called logical. (This property is called \permutation invariant" in Jonsson [Jo86].) In this work we concentrate on that version of the above quoted problem of Henkin and Monk in which all operations of the set algebras are required to be logical.6 This condition admits the following equivalent formulation. The class of representable algebras can be obtained in the form Alg(L) for some logic L such that, in the model theory of L, isomorphic models satisfy the same formulas | the latter condition is one of the basic axioms of abstract model theory. A solution to the truly nite axiomatization version of the nitization problem was given in Sain [Sa87a] part of which is revised as [Sa87] and abstracted in [Sa92] and [Sa94]. These papers give a complete solution to the case of rst order logic without equality, while the solution there to logic with equality is slightly weaker. Much of the e ort in the just quoted papers goes into ensuring that the number of axioms is strictly nite (as opposed to a nite schema). Therefore the idea comes up that if we require only a nite schema axiomatization (as opposed to \truly nite") then, perhaps, we could achieve substantial improvements in other directions such as simplifying the axioms. Indeed, after [Sa87a] was presented at the Algebraic Logic conference in Asilomar, 1987, both Leon Henkin and J. Donald Monk suggested looking for a solution which, instead of a truly nite set of axioms, provides only a nite schema of axioms, but in such a way that this schema would be simple and easy to work with. This is one of the things we intend to do in the present paper. 7 The question of comparing our notion of a schema with the notion used in Keisler's completeness theorem and in the Daigneault{Monk representation theorem for polyadic algebras (without equality) comes up at this point (see [HMTII] 5.4). We will make such a comparison using results of [NS96] and [Sa95] in Subsection 2.6 at the end of this section. 5 Tarski mentions this requirement already in his 1941 paper on the calculus of relations. See also next footnote. 6 This condition goes back to Lindenbaum{Tarski [LT36], and was extensively discussed in Tarski [Ta87] where it was suggested that the condition is essential for investigations of truly logical nature. Further considerations for necessity of requiring all the operations to be logical (i.e. permutation invariant) are in [Sh91] and [vB96] 3.3.2 (pp.67{68). It was proved e.g. in Biro [Bi89] that permutation invariance is a necessary condition in the nitization problem. Cf. also Nemeti [N91]. 7 A related open problem is to simplify the axiomatization given in the original papers (e.g. in [Sa87]).
4
A further aim of this paper is to improve the positive results to a tighter or more orthodox class of set algebras. This tighter class is denoted below as CdfG . To rst order logic without equality, we will associate a fairly simple and powerful class CdfG of set algebras. A CdfG algebra A is a subalgebra of the Boolean algebra of subsets of the Cartesian space ! U expanded with some extra operations corresponding to quanti cation and substitution8 . We will axiomatize CdfG by a relatively simple nite schema of quasi-equations (cf. Theorem 2.5). A modi ed version of this class will receive an equational axiomatization. It remains an open problem to simplify these axiomatizations. To the same logic we will also associate a slightly bigger class GwdfG of set algebras. We will be able to prove more results about this second class (and also we will be able to simplify the axioms for the latter). CdfG will meet all the requirements of the (schema version of the) nitization problem quoted way above. E.g., its operations will be logical and natural set theoretic operations on relations. The same applies to GwdfG . To rst order logic with equality, we will associate a class Cssuc;pred of set algebras, which has the following positive properties. The equational class V generated by Cssuc;pred is axiomatizable by a nite schema of equations, which is even simpler than the one we give to logic without equality. Further, Cssuc;pred is an expansion of cylindric set algebras with some extra operations. These extra operations are again logical and simple set theoretic ones, just as in the previous case. Now, our nite schema axiomatizable V has the property that if we omit the extra operations from its members then we get exactly the class RCA! of representable cylindric algebras. Let us see what this means in more detail. To the natural algebraic counterpart RCA! of rst order logic with equality, we add simple logical operations in such a way that, in the language of the so expanded class, we can write up a nite schema of axioms which completely describes the original operations of RCA! , but not necessarily the new operations themselves. More concretely, any model A of our nite schema is representable as a true set algebra in the following sense. A is isomorphic to some B, where the elements of B are real relations, and all the old cylindric algebraic operations are the usual set theoretic ones on B, but perhaps the new operations of B remain \abstract". At this point it might be of interest to note that, in the book [TaG87] (on page 62), the nitization problem was formulated in this weaker sense which we just described in connection with logic with equality. That is, Tarski and Givant ask for expanding the existing algebras with new logical operations in a way that there would be a nite set of axioms in the expanded language which 8 These algebras are often called \square algebras".
5
describes the old operations completely, but not necessarily the new ones. For a comparison of the two versions of the nitization problem see the series of remarks in [N93] beginning with Remark 2. Summing up, we can give a solution to the more \ambitious" formulation of the schema version of the nitization problem for logic without equality, while for logic with equality we could solve only the above outlined Tarski-Givant style formulation of the nitization problem. When de ning the nite schema axiomatizable ( nitizable for short) algebraic counterparts GwdfG; Cssuc;pred of rst order logic, one has a certain freedom of choice concerning exactly how these algebras will look like. We will explore this freedom of choice in a metatheory, asking ourselves the question which of these choices of GwdfG ; Cssuc;pred are workable, and which are already non- nitizable. We would like to note that our approach to the case with equality is strongly related to William Craig's pioneering work summarized, e.g., in his book [Cr74]. Actually, our work can be viewed as a continuation of the approach initiated by Craig. *** The formulation of the results mentioned above together with the basic de nitions can be found in Section 2. In detail, Subsection 2.1 contains preliminary material for the rest of the paper, while Subsections 2.2 and 2.3 are devoted to the nitization of rst order logic without equality and with equality, respectively. Subsection 2.5 concerns applications to `pure' logic. The proofs are given in Section 3. At the end of the paper there are further historical remarks. Any proof which would rely on references not available to the reader (or would seem incomplete for any other reason) is available with all such missing data added from the authors.
2 Results
2.1 Preliminaries
Throughout this paper, ! denotes the set of all natural numbers. If A and B are arbitrary sets then A B denotes the set of all functions from A into B . Thus, for any set U , ! U denotes the set of all !{sequences over U . We often call the elements of ! ! transformations of !. A function f 2 ! ! is called a nite transformation of ! if f (i) = i for all but nitely many i 2 !. Examples of nite transformations of ! are transpositions and replacements, which are de ned as follows. For each i; j 2 !, the transposition [i; j ] interchanges i and j and leaves all the other elements of ! xed; the replacement [i=j ] maps i to 6
j and leaves all the other elements of ! xed. It is easy to see that the set of all nite transformations of ! forms a semigroup with function composition as the semigroup operation. Clearly, this semigroup is generated by the set of all replacements and transpositions. Examples of transformations of ! that are not nite are the successor suc and the predecessor pred de ned as follows. For every i 2 !, suc(i) = i + 1 and i ? 1 if i 6= 0 . pred(i) = i if i = 0 If H is a set then H denotes the set of sequences of elements of H . If T is a subset of ! ! then [T ] denotes the subsemigroup of h! !; i generated by T . For any class K of similar algebras, IK , HK , SK , PK and UpK denote, respectively, the classes of isomorphic copies, homomorphic images, subalgebras, isomorphic copies of direct products and isomorphic copies of ultraproducts of members of K . SirK denotes the class of subdirectly irreducible members of K. Let t and t0 be similarity types with t0 t. For any class K of algebras of similarity type t, Rdt0 K denotes the class of algebras of similarity type t0 de ned as follows. A 2 Rdt0 K if and only if A is the t0 {reduct of some algebra in K . Informally speaking, we obtain the t0 {reduct of an algebra by deleting those operations of it that are not in t0 . For any set of rst order formulas of similarity type t, Modt () denotes the collection of those models (of type t) in which is valid. When t is known from context, we will often write Mod() instead of Modt (). For a class of algebras K , Eq(K ) denotes the set of equations valid in K while Qeq(K ) is the set of valid quasi-equations of K . If f is a function and U is a subset of its domain then f dU denotes the restriction of f to U . For the pair of sets U; V , U ! V expresses the relation that U is a nite subset of V . As we outlined in the Introduction, we will introduce several (new) kinds of set algebras. These algebras will be de ned to be Boolean algebras of relations augmented with some extra operations. The greatest element of a Boolean algebra (with perhaps some extra operations) is called its unit. Therefore, when speaking about a class K of special Boolean algebras, then it is meaningful to talk about the class of unit elements of members of K . These will be called K {units. Of these classes of set algebras, our rst theorem (Theorem 2.5) will involve only the most important one called CdfG . Actually, one of the main results of this paper is Theorem 2.5 (i), (ii). To understand that result it is enough to read only a small fraction of the de nitions below. Namely, after reading De nition 2.1 (i), the CdfH part of De nition 2.2 and De nition 2.4 the reader can go directly to read Theorem 2.5. 7
df-unit
De nition 2.1 ([HMTII] Def.3.1.2)
Let U be an arbitrary set. (i). The set ! U is called the Cartesian space with base U (and dimension !). (ii). By a generalized Cartesian space we mean a disjoint union of Cartesian spaces. That is, a set [ is called a generalized Cartesian space (of dimension !) if it has the form ! Ui , where I is a set, and for all distinct i; j 2 I , i2I
Ui \ Uj = ;. (iii). For every p 2 ! U , we set ! U (p) = fq 2 ! U :
fi 2 ! : q(i) 6= p(i)g is niteg;
and we call ! U (p) the weak Cartesian space (of dimension !) determined by p. U is called the base of ! U (p) . (iv). By a generalized weak Cartesian space or Gws! {unit we mean a disjoint union of weak Cartesian spaces. That is, a set is called [ a generalized weak Cartesian space of dimension ! if it has the form ! Ui(pi ) for some sets
i2I I; Ui and functions pi 2 ! Ui (i 2 I ) such that for all distinct i; j 2 I , ! U (pi ) \ ! U (pj ) = ;. i j [ (v). We say that a Gws! {unit ! Ui(pi ) is normal if for all distinct i; j 2 I , i2I Ui and Uj are either disjoint or equal. [ (vi). Finally, we say that a Gws! {unit ! Ui(pi ) is compressed if Ui = Uj for i2I all i; j 2 I .
2.2 Finitizing algebras of rst order logic without equality
De nition 2.2 Suppose that H is a xed subset of ! !. An algebra is called a full generalized weak H {cylindric set algebra with unit V if it has the form
hP (V ); [; ?; s ; ck i 2H;k2! ; where
V is a Gws! {unit, for every 2 H and x V , s (x) = fq 2 V : q 2 xg, 8
d-igwsg
s (V ) = V whenever 2 H , ck is the k-th cylindri cation, that is, for any x V , ck (x) = fq 2 V : (9q0 2 x)(8i 2 !) i =6 k ) q(i) = q0(i)g: If A is a full generalized weak H {cylindric set algebra with unit V , then A will often be denoted by P (V ), if H is known from context. By a generalized weak H {cylindric set algebra (or brie y, by a GwdfH ) we mean a subalgebra of a full generalized weak H {cylindric set algebra. The class of all generalized weak H {cylindric set algebras is denoted by GwdfH 9 . The similarity type of GwdfH is denoted by tH . The classes of all algebras of GwdfH , whose unit elements are a Cartesian space, a generalized Cartesian space, a normal Gws! {unit or a compressed Gws! {unit, are denoted by CdfH , GdfH , GwdfHnm or GwdfHcm respectively.
De nition 2.3 Let H ! !, V be a generalized Cartesian space10, x V . We de ne the operator c(!) : P (V ) ! P (V ) as
d-111
c(!)x def = fs 2 V : (9q 2 c0 x) s(0) = q(0)g: A = hBec; c(!) i is called GdfHec i B is a GdfH with unit V . An A 2 GdfHec is
called CdfH if its unit element is a Cartesian space.
To help the intuition we point out that IGdfHec = SPCdfHec while in CdfHec c(!)x = 1 i x > 0 and c(!)0 = 0. Adding the new operator c(!) to GdfH amounts to adding the formation of existential closure ' ! 9x' to the logic being algebraized (where x contains all the variables occuring in '). The abbreviation `ec' in GdfHec (and in CdfHec) refers to this fact.
De nition 2.4 Throughout this paper, G denotes the following set.
d-g
G def = f[i=j ] : i; j 2 !g [ fsuc; predg: The following theorem summarizes the main results of this section. 9 Gwdf refers to the name `generalized weak diagonal free cylindric set algebra' used in
[HMTI] to denote the H -free reduct of our class. 10 The de nition makes perfect sense for the case when V is only a generalized weak Cartesian space (or even any subset of ! U for some set U ). For brevity, we will not discuss here the more general cases.
9
csicsa1
Theorem 2.5 (i). ICdfG is a nite schema axiomatizable quasi-variety. That is, there is a nite schema QxG 11 of quasi-equations such that ICdfG = Mod(QxG) :
(ii). IGdfGec is a nite schema axiomatizable discriminator variety. (iii). HCdfG is a variety axiomatized by a nite set AxG 12 of equation schemas. Thus, AxG axiomatizes Eq(CdfG ) i.e. CdfG j= e i AxG ` e; for any equation e in the language of CdfG . The statements of this theorem are restated in Theorems 2.7, 2.15, 2.16 below in a more detailed form. In the following part of this section we present the nite sets of equations and quasi-equations that axiomatize HCdfG , ICdfG and IGdfGec. Further, the related weak classes are examined and a detailed characterization is given. (Theorem 2.7) Finally, we show that many of the results remain valid if we replace the set G by an arbitrary set H of transformations provided that H satis es certain properties. (Theorems 2.11, 2.19, 2.20) De nition 2.6 We de ne the nite set AxG of axiom schemas to be the union of sets (1){(6) below. (1). A nite set of equations axiomatizing Boolean algebras. (2). For every 2 G, the following axioms stating that s is a Boolean homomorphism: s (x _ y) = s (x) _ s (y); s (?x) = ?s (x): (3). The axioms governing s[i=j] 's: for all distinct i; j; k 2 !, s[j=i] s[i=j] x = s[j=i] x; s[j=k] s[i=j] x = s[j=k] s[i=k] x: (4). The axioms describing the connections between s[i=j] 's, ssuc and spred: spredssucx = x; ssucspredx = s[0=1]x; ssucs[i=j] x = s[suc(i)=suc(j)] ssucx for all distinct i; j 2 !. (5). Two axiom schemas about ci and the Boolean operations: for all i 2 !, x ci x, ci (x _ y) = ci x _ ci y: (6). The axiom schemas describing the connection between s[i=j] 's and ck 's: for all distinct i; j; k 2 !, s[i=j] ci x = cix; cis[i=j] x = s[i=j] x; ck s[i=j] x = s[i=j] ck x: 11 QxG is presented in De nition 2.14 below. 12 AxG is presented in De nition 2.6 below.
10
d-ax
Theorem 2.7 (characterization of IGwdfG and related classes)
th-main
Let G and AxG be as in De nitions 2.4 and 2.6. Then statements (1){(4) below hold. (1). IGwdfG is axiomatizable by the nite set AxG of equation schemas i.e. IGwdfG = Mod(AxG). (2). IGwdfG is a variety. (3). IGwdfG = IGwdfGnm = IGwdfGcm = HCdfG . (4). The set Eq(CdfG) of equations is axiomatizable by the nite set AxG of equation schemas i.e.
CdfG j= e i AxG ` e; for any equation e in the language of CdfG .
Proof (2) is a corollary of (1) while (1) is proved in Subsections 3.1 { 3.3.
To prove (3) we note that statement (3) is a special case of Theorem 2.11 (ii), (iii). To see that the later is applicable, we need to show that the successor, the predecessor and all the substitutions are bounded transformations of ! and they have right quasi-inverses in [G]. (Consult De nition 2.9 for the notion of bounded, quasi-bounded and quasi-invertible transformation.) It easy to verify that the given transformations have the required properties using facts as suc pred(i) = pred suc(i) = i whenever i > 0. Finally, (4) follows from (3) and (1).
Later, in Remark 2.22, we show that for a slightly modi ed version of IGwdfG (where only the 0-th cylindri cation is a basic operation, the other cylindri cations are terms of the reduced language) similar results can be achieved. In the proof of the previous theorem we referred to a metatheorem (2.11). There we show that if H satis es certain requirements then the classes IGwdfH and IGwdfHcm actually coincide with IGwdfHnm.
Notation 2.8 We will often deal with pairs of sequences that agree on all but
nt-approx
nitely many coordinates. If q1 and q2 are sequences then
q1 q2 i jfi 2 ! : q1 (i) 6= q2 (i)gj < !: Clearly, is an equivalence relation.
De nition 2.9 Let f; g 2 ! ! be arbitrary. 11
d-bounded
f is said to be bounded if it does not send in nitely many elements of ! into the same place, that is, if jf ?1(i)j < ! for all i 2 !. f is said to be quasi-bounded if there is a k 2 ! for which jf ?1 (f (k))j < !. g is called the right (left) quasi-inverse of f if f g Id (g f Id), where Id is the identity transformation of !. f is said to be right (left) quasi-invertible in H ! ! if it has a left (right) quasi-inverse in H .
De nition 2.10 (rich semigroup) Suppose that i; j; k 2 !, A !, f; g 2 ! !. Then f [i=j ] and f [A=g] are transformation of ! de ned as
f [i=j ](k) =
j if k = i f (k) otherwise
f [A=g](k) =
d-rich
g(k) if k 2 A f (k) otherwise.
(Note that f [i=j ] and f [i=j ] are di erent transformations.)13 A semigroup Q ! ! is called rich if it satis es the following two conditions: (8i; j 2 !)(8f 2 Q) f [i=j ] 2 Q, (9s^; p^ 2 Q)(8f 2 Q) p^ s^ = Id ^ Rng(^s) 6= ! ^ (^s f p^)[(! ? Rng(^s))=Id] 2 Q.
Theorem 2.11 Let H be an arbitrary set of transformations. (i). IGwdfHnm = HSPCdfH is a variety. Thus Eq(CdfH ) = Eq(GwdfHnm). (ii). If for all 2 H , is bounded then IGwdfHnm = IGwdfHcm = HCdfH : (iii). If for all 2 H , is quasi-bounded and right quasi-invertible in [H ], then
IGwdfH = IGwdfHnm: (iv). Assume that [H ] is a nite schema presented14 rich semigroup. Then, IGwdf nm is axiomatizable by a nite schema of equations.
(i) and (iv) are quoted from [Sa92] (Theorem 2.1 (2)), the proof of (ii) and (iii) can be found in Subsection 3.3. The example below shows that the requirement given in Theorem 2.11 (iii) cannot be omitted. In some cases Eq(GwdfH ) 6= Eq(GwdfHnm ). 12
th-nc
Example 2.12 Let doub 2 ! ! be de ned by doub(i) = 2i for every i 2 !, let H = fdoubg and let e be the equation c0 sdoub x = sdoubc0 x. Then GwdfHnm j= e but GwdfH 6j= e: Proof Let V = ! 2(h0;1;0;1;:::i) [ ! 3(h0;0;0;0;:::i). Then P (V ) 2 GwdfH n IGwdfHnm since e fails in P (V ) (choose x = fh2; 0; 0; 0; : : :ig) but e is satis ed in every algebra of IGwdfHnm. We turn our attention to the study of classes of algebras with square (Cartesian space) unit elements. We prove that the class ICdfG is a nite schema axiomatizable quasi-variety and is strictly smaller than IGwdfG .
ex0
Example 2.13 Let q be the quasi-equation spredx = ?x ) 0 = 1. Then15 GdfG j= q but IGwdfG 6j= q: Proof In a G{cylindric generalized set algebra we can pick a constant sequence s in the universe. For any element x of the algebra, the chosen s is in x if and only if it is in spredx. It implies that q is in the symmetric di erence of ?x and spredx so spredx =6 ?x. On the other hand, if we put V = ! 2(h1;0;1;0;:::i) [ ! 2(h0;1;0;1;:::i) then e fails in P (V ). (Put x to be ! 2(h0;1;0;1;:::i) and for that x, spredx = ?x.)
ex1
De nition 2.14 (i). Let n 2 !. We de ne c(n)x def = c0 c1 : : : cn?1 x. Furdef ther, sn-pred x = spreds[0=1]s[1=2] : : : s[n?1=n] x. Hence, sn-pred is a term
d-qx1
function16 .
13 f [i=j ] can be distinguished from f [A=g] by noticing that always jgj ! > i. 14 We use the notion of a nite schema presented semigroup in the usual sense. For com-
pleteness we will recall this concept in De nition 3.7 in the `Proofs' section. 15 For completeness we note connections of Example 2.13 and the theory of polyadic algebras (PA! 's) and PA's with equality (PEA! 's) without recalling these two classes. We use the notation of [HMTII]. Although AxG 6j= q, we have PA! j= q (e.g. by the Daigneault{Monk theorem). On the other hand, there are equations e such that PEA! 6j= e but RPEA! j= e, cf. Nemeti-Sagi [NS96]. 16 In set algebras, the term function sn-pred coincides with s for = h0; 1; : : : ; n ? 1; n ? 1; n; n + 1; : : :i, that is, n if i < n (i) = (predd! ? n) [ (Iddn) = ipred(i) otherwise.
13
(ii). We de ne QxG to be the following schema of quasi-equation.
[
im
c(ni)xi = 1 ^
for n0 ; n1 ; : : : ; nm ; m 2 !.
Theorem 2.15 Proof
17
^
im
sni-predxi ?xi ) 0 = 1;
cj1
ICdfG = Mod(AxG [ QxG):
The general Theorem 2.19 should be applied to H = G. Qx can be simpli ed to QxG and Theorem 2.7 implies that Eq(IGwdfG ) follows from AxG . It is left to the reader to show the rst implication.
We conjecture that QxG can be considerably simpli ed. Cf. e.g. the de nition of Ax2 and proof of Proposition 1 in Andreka [A91] part II. ec Theorem 2.16 IGdf G is a nite schema axiomatizable discriminator variety. ec Furthermore, IGdfG = Mod() where is AxG together with the following set of axiom schemas: (i). axioms stating that c(!) is a complemented closure operator, i.e. x c(!) x, c(!)(x [ y) = c(!)x [ c(!)y, c(!)c(!)x = c(!)x, c(!)?c(!)x = ?c(!)x, (ii). axiom schemas stating that c(!) majorizes all the other operators, i.e. c(!)x f (x) for all f 2 fci; s : i 2 !; 2 Gg, (iii). axioms corresponding to[QxG : for all n0 ; n1 ; : : : ; nm; m 2 !, [ ?c(!) ?( c(ni ) xi ) c(!) (xi \ sni -pred(xi )).
ICdf ec
im
t-nemeti
im
ec G is the class of subdirectly irreducible members of IGdfG .
Theorem 2.16 is a special case of the general Theorem 2.20. Open question 2.17 Find simpler axiom schemas aximatizing ICdfG and IGdfecG, respectively. Below we show that ICdfH is a quasi-variety, provided that H satis es some properties. Furthermore, we give a set Qx of quasi-equations that axiomatize ICdfH over IGwdfH . Results 2.19 and 2.20 were obtained jointly with Istvan Nemeti. De nition 2.18 (i). Let n 2 !, ? = f 0; : : : ; n?1g !. We recall from [HMTI] that c(?) x def = c 0 : : : c n?1 x.
17 In [Sa87] a simple proof is given to the fact that ICdfG is closed under SPUp, i.e. it forms a quasi-variety. Finite schema axiomatizability is not proved there.
14
ques1
d-qx2
(ii). We de ne Qx to be
[
im
c(?i)xi = 1 ^
^ \ im 2Hi
s (x) = 0 ) 0 = 1;
for m 2 !; ?0 ; : : : ; ?n?1 ! !, Hi ! [H ] such that ( 2 Hi ) d? = i Idd? ). i
Theorem 2.19 Let H ! ! such that [H ] contains all nite transformations
t-csax
and every element of H is bounded, left and right quasi-invertible in H . Then (1). ICdfH is a quasi-variety, (2). ICdfH is axiomatized by Qx over GwdfH , that is, Qx [ Eq(GwdfH ) axiomatizes ICdfH .
(1) is a consequence of (2) while (2) is proved in Subsection 3.4.
Theorem 2.20 Assume that H ! ! is such that ICdfH is a quasi-variety. Then IGdfHec is a discriminator variety whose subdirectly irreducible members are exactly the elements of ICdfHec . If all the assumptions of Theorem 2.19 hold for H then IGdfHec is axiomatized over IGwdfH by the axioms (i) and (ii) of Theorem 2.16 together with (iii'). axioms corresponding to Qx: for all m 2 !; ?0; : : : ; ?n?1 ! !, Hi ! [H ] such that ( 2 Hi ) d? = Idd? ) we state [ [ i \ i ?c(!) ?( c(?i ) xi ) c(!) ( s x). im
im
2Hi
Proof Let be theecset equations validecin GdfHec, that is, = Eq(GdfHec). We want to prove IGdfH = Mod(). IGdfH Mod() is obvious. For proving the other direction, it is enough to show that
Sir(Mod()) ICdfHec IGdfHec: Therefore, assume A 2 Sir(Mod()). A j= x 6= 0 ! c(!) x = 1 can be veri ed as follows. Suppose for contradiction that c(!) x = 6 1 for some element x of A. ec
Since axioms (i) and (ii) certainly hold in GdfH , they are in . But using axioms (i) and (ii) we can prove that the relativization with x and that with ?x decomposes A to the ^ subdirect product of the two images. Let A = hB; c(!) i. If i = i ! = is a quasi-equation valid in CdfH iFrom the main result of Thompson [To93] we know that [1=0]n 0 [1=0]m = [1=0]n 00 [1=0]m imply ` [0=1]n 0 [0=1]m = [0=1]n 00 [0=1]m. Collecting all the information we have: ` sucn predm = sucn predm . But this implies ` predn sucn predm sucm = predn sucn predm sucm . Using (3.1) again we can conclude ` = and this completes the proof. The set of de ning equations listed in 3.8 is not independent. Below we show that equations (3.4), (3.5), (B6), (B7) are implied by the others.
Claim 3.9 (3.1),(3.2),(3.3),(B1),(B3),(B4),(B5)`(B6) Proof First we prove that the listed B-axioms imply [i=j ][j=k][k=i][i=j ][a=b] = [i=k][k=j ][j=i][a=b] where all i; j; k; a; b are distinct: [i=j ][j=k][k=i][i=j ][a=b]
B=1
B=4
B=1
B 3;B 4
= = B=3
B 5;B 3;B 4;B 1
26
[i=j ][j=k][k=i][i=j ][a=j ][a=b] [i=j ][j=k][k=i][a=j ][i=j ][a=b] [i=j ][j=k][k=i][a=i][a=j ][i=j ][a=b] [i=j ][j=k][a=i][k=a][a=j ][i=j ][a=b] [a=j ][i=j ][j=k][i=j ][k=a][a=b] [a=j ][i=j ][j=k][i=k][k=a][a=b]
cl-b7
B 4;B 1;B 3
=
B=5
B 3;B 4;B 5
= B 1;B 5 =
[a=j ][i=k][j=i][k=a][a=b] [i=k][a=j ][k=a][j=i][a=b] [i=k][k=j ][a=j ][a=b][j=i] [i=k][k=j ][j=i][a=b]:
Now we can compute: [i=j ][j=k][k=i][i=j ] 3=:1 pred suc[i=j ][j=k][k=i][i=j ] 3:3;3=:1;3:2 pred[i + 1=j + 1][j + 1=k + 1][k + 1=i + 1][i + 1=j + 1][0=1]suc = pred[i + 1=k + 1][k + 1=j + 1][j + 1=i + 1][0=1]suc 3:3;3=:1;3:2 [i=k][k=j ][j=i].
Claim 3.10 (3.1),(3.2),(3.3),(B1),(B3),(B4),(B5)`(B7) Proof First we prove that the listed B-axioms imply
cl-b8
[i=k][k=l][l=j ][j=i][i=l][a=b] = [i=j ][j=k][k=l][l=i][a=b] where all i; j; k; l; a; b are distinct: [i=k][k=l][l=j ][j=i][i=l][a=b]
B 1;B 4
= = B 4;B 5;B 3 = B 4;B 3 = B 5;B 1 = B=5 B 3;B 4;B 5;B 1 = B 1;B 5 = B 1;B 4;B 3
[i=k][k=l][l=j ][j=i][a=l][i=l][a=b] [i=k][k=l][l=j ][a=i][j=a][a=l][i=l][a=b] [a=k][i=k][k=l][l=j ][i=l][j=a][a=b] [a=k][i=k][k=l][i=j ][l=i][j=a][a=b] [a=k][i=j ][k=l][l=i][j=a][a=b] [i=j ][a=k][j=a][k=l][l=i][a=b] [i=j ][j=k][a=k][a=j ][k=l][l=i][a=b] [i=j ][j=k][k=l][l=i][a=b]:
Now we can compute: [i=k][k=l][l=j ][j=i][i=l] 3=:1 pred suc[i=k][k=l][l=j ][j=i][i=l] 3:3;3=:1;3:2 pred[i + 1=k + 1][k + 1=l + 1][l + 1=j + 1][j + 1=i + 1][i + 1=l + 1][0=1]suc = pred[i + 1=j + 1][j + 1=k + 1][k + 1=l + 1][l + 1=i + 1][0=1]suc 3:3;3=:1;3:2 [i=j ][j=k][k=l][l=i].
Claim 3.11 (3.1),(3.2),(3.3),(B3),(B4)`(3.5) 27
cl-35
Proof [0=j ]pred 3=:1 pred suc[0=j ]pred 3=:3 pred[1=j + 1]suc pred 3=:2
pred[1=j + 1][0=1] B3=;B4 pred[0=j + 1][1=j + 1]
Claim 3.12 (3.1),(3.2),(3.3),(B4),(B5)`(3.4)
cl-34
Proof [i=j ]pred 3=:1 pred suc[i=j ]pred 3=:3 pred[i + 1=j + 1]suc pred 3=:2
pred[i + 1=j + 1][0=1] B4orB = 5 pred[0=1][i + 1=j + 1] 3=:2 pred suc pred[i + 1=j + 1] 3=:1 pred[i + 1=j + 1] We quote an important result of Pinter.
Claim 3.13 (Lemma 2.2 of [P73]) For every distinct i; j; k; l 2 ! we have AxG ` s[i=k] s[i=j] = s[i=j] ; AxG ` s[k=j] s[i=j] = s[i=j] s[k=j] ; AxG ` s[k=l] s[i=j] = s[i=j] s[k=l] : We will need the following technical observation that originates from Sandor Csizmazia:
Lemma 3.14 AxG together with the equations (*3), (*4), (*5), (*1) without ci cj x = cj cix implies that for any 2 f[i=j ] j i; j 1g [ f[1=0]suc; pred[1=2]g s c0x = c0s x Proof There is nothing to prove in s[i=j] c0x = c0s[i=j] x (i; j 1) since it is
included in AxG . It was formulated in the lemma only for further applications. To prove the other two statements we will need the following claim quoted from Pinter [P73]:
z
B
}|
{
cl-p
lm-uj7
cl-ci
Claim 3.15 cix = min fy j y = s[i=j] y; y xg Proof ci x 2 B since s[i=j] ci x = ci x and cix x that is (5) and (6) of AxG. 1 For any y 2 B y ci x since ci x ci y = ci s[i=j] y =7 s[i=j] y = y. Further observations are needed.
cl-defci
Claim 3.16 cix = s[0;i]c0 s[0;i]x Proof Using the previous claim, we have to prove that s[0;i]c0 s[0;i]x 2 B def = fy j y = s[i=j] y; y xg and y 2 B ) y s[0;i] c0 s[0;i] x. s[i=j] s[0;i]c0s[0;i]x =3 s[0;i]s[0=j] c0s[0;i]x =6 s[0;i]c0s[0;i]x: 28
(5) implies c0 s[0;i] x s[0;i] x, using (2) we have s[0;i] c0 s[0;i] x s[0;i] s[0;i] x =4 x. 1;2 y 2 B ) y x ) s[0;i] c0s[0;i]x s[0;i]c0s[0;i]y y=2B s[0;i]c0s[0;i]s[i=j] y =3 s[0;i]c0s[0=j] s[0;i]y =6 s[0;i] s[0=j] s[0;i]y =3 s[0=j] y y=2B y: cl-c1s01
Claim 3.17 c0s[1=0]x = c1s[0=1]x Proof c0s[1=0]x =3 c0s[1=2]s[1=0]x =6 s[1=2]c0 s[1=0]x =3 s[1=0]s[1=2]c0s[1=0]x =6 s[1=0]c0s[1=2]s[1=0]x =3 s[1=0]c0s[1=0]x =3 s[0;1]s[0=1]c0s[0;1]s[0=1]x =6 s[0;1]c0s[0;1]s[0=1]x Cl:=3:16 c1s[0=1]x Claim 3.18 ci+1ssucx = ssucci x Proof Let B1 = fy j s[i+1=i+2] y = y; y ssucxg; B12 = fy j s[i+1=i+2] y = s[0=1] y = y; y ssucxg; B21 = fssuc y j s[i=i+1] y = y; y xg; B2 = fy j s[i=i+1] y = y; y xg: The following statements prove this claim: I ci+1 ssuc x = minB1 ;
. II minB1 = minB12 ; III B12 = B21 ; IV minB21 = ssuc minB2 ; V minB2 = ssuc ci x: I, V follow from Claim 3.15, IV from the observation that y 2 B2 $ ssucy 2 B21 . Similarly to the proof of the previous claim, in II we need to show that B12 B1 (trivial) and (8y 2 B1 ) ? c0 (?y) 2 B12 ^ ?c0 (?y) y. So suppose y 2 B1 . Then s[i+1=i+2] (?c0 (?y)) Ax = ?c0 (?s[i+1=i+2] y) = ?c0 (?y), s[0=1](?c0(?y)) = ?s[0=1]c0(?y) = ?c0 (?y) and y ssuc x ) ?c0 (?y) ?c0 (?ssucx) = ?c0 ssuc(?x) = ?ssuc (?x) = ssucx. In * we used that ssuc x = ssucspredssucx = s[0=1] ssucx = c0 s[0=1] ssuc x = c0 ssucx. Turning to III, suppose that y 2 B12 . Then y = s[0=1] y = ssuc (spredy). s[i=i+1](spredy) = spredssucs[i=i+1](spredy) = spreds[i+1=i+2] ssucspredy = spreds[i+1=i+2] s[0=1]y = spredy, y ssuc x ) spredy spredssucx = x. On the other hand if we suppose that ssucy 2 B21 then s[i+1=i+2] ssucy = ssucs[i=i+1] y = ssucy, s[0=1]ssucy = ssucs[ssuc=y] = ssucy and y x ) ssucy ssuc x.
Now we can prove the two statements formulated in the lemma: c0 s[1=0]ssucx = ssucc0 x =6 ssucs[0=1]c0x =4 s[1=2]ssucc0x =3 29
cl-cisuc
=3 s[1=0] s[1=2] ssucc0 x = s[1=0] ssuc c0 x, c0 spreds[1=2]x =4 spredssucc0 spreds[1=2]x = spredc0s[0=1]ssucspreds[1=2]x = = spredc0 s[1=2] x = spreds[1=2] c0 x. In * we used c0 s[1=0] ssucx Cl:=3:17 c1 s[0=1] ssucx Ax = c1 ssucspredssucx Ax = Ax Cl: 3:18 = c1 ssuc x = ssucc0 x.
Proof of Theorem 2.7 (1) Using the result of the previous subsection stated in Corollary 3.6 we need to prove only that AxG `(*1){(*7) and AxG ` ci x = s[0;i]c0s[0;i]x. (*2) is a trivial consequence of (2). (3) and (4) together imply (*3) and (*4) since if A is an algebra of IGwdfG and A is the semigroup of all polynomials of A, expanded with the constant symbols [i=j ]; suc; pred evaluated into the polynomials s[i=j] ; ssuc; spred then A is a model of (3.1){(3.6) ((3) and (4) is the relevant part of the set of equations (3.1){(3.7) | see Claim 3.9{3.13 | written in the language of A ) and so = forces A j= = and that is s x = s x. (*7) is included in (6). To prove (*5), suppose and are in G and and agree everywhere but in i. Then [i=i + 1] and [i=i + 1] are equal everywhere. s ci x =6 s s[i=i+1]ci x = s [i=i+1]ci x = = s [i=i+1] ci x = s s[i=i+1] ci x =6 s ci x In (*1) x ci x, ci (x _ y) = ci x _ ci y follow from (2) and (5). ci ci x = ci x and ci (?ci x) = ?ci x is a consequence of (2) and (6): ci ci x = ci s[i=i+1] ci x = s[i=i+1]ci x = ci x and ci(?cix) = ci (?s[i=i+1] cix) = ci s[i=i+1] (?ci x) = = s[i=i+1] (?ci x) = ?s[i=i+1] ci x = ?ci x To prove (*6) we will need the following observation: If T0 = ff 2 [G] j f ?1 (0) = f0gg and G0 = f[i=j ] j i; j 1g [ f[1=0]suc; pred[1=2]g then T0 is generated by the meanings of the elements of G0 . To verify this statement de ne the function 0 mapping [G] into itself
f 0 = (suc f pred)[0=0] (Since [G] is rich f 0 2 [G].) It is easy to check that 0 is a homomorphism ((f g)0 = f 0 g0 ), furthermore, the 0 image of [G] is T0 (if f 2 [G] then f 0?1 (0) = f0g so f 0 2 T0 and if g 2 T0 then (pred g suc)0 = g). Finally, we note that the set of transformations named by the elements of G0 coincide with the 0 image of the set of those transformations that are named by the elements of G. We note that for any 2 G0 , s c0 x = c0 s x. This follows from Lemma 3.14 since all the extra equations listed in the formulation of the lemma are 30
consequences of AxG , as it was proved up till this point. In passing, we mention that the same reasoning shows that Claim 3.16 proves that the de nition of ci (i > 0) is implied by AxG . Turning our attention again to (*6), suppose 2 G , ?1 (j ) = fig. Then f = [j; 0] [i; 0] 2 T0 . Using the previous observation, f is a composition of functions named by elements of G0 say, f = 1 : : : n . (3), (4) and (6) imply that s[j;0] [i;0]c0x = s1:::n c0x = s1 : : : sn c0x = c0s1 : : : sn x = c0 s[j;0] [i;0]x: Finally we derive
s ci x =y s s[i;0]c0s[i;0]x = s[j;0]s[j;0] [i;0]c0s[i;0]x = = s[j;0] c0 s[j;0] [i;0]s[i;0] x =y cj s x: (In y we used that the de nition of ci and cj is already proven from AxG .) It is left to prove ci cj x = cj ci x in (*1). For that we need x y ) ci x ci y and ci cj ci x = cj ci x. ci cj ci x =6 ci cj s[i=k] ci x =6 ci s[i=k] cj ci x =6 s[i=k] cj ci x =6 cj s[i=k] ci x =6 cj ci x. Now ci cj x ci cj ci x = cj ci x and symmetrically cj ci x ci cj x giving cj ci x = ci cj x. In the previous two subsections we distinguished the symbol from the transformation denoted by it. This distinction is not necessary anymore since axioms (*3), (*4) of De nition 3.3 hold in all algebras studied hereafter. So we return to the original notation where denotes both the symbol and the corresponding transformation hoping that context will help.
3.3 Theorem 2.11: compressing the unit element
Proof of Theorem 2.11cm (ii) IGwdfHcm = HCdfH can be veri ed as follows. Suppose that A 2 GwdfH . Put V to be the unit element, A the universe of A.
Since V is a compressed Gws! {unit, every weak space in V has the same base, say U . Put B = fx ! U : x \ V 2 Ag. Clearly, B def = hB; [; ?; s ; ck i 2H;k2! is in CdfH and the relativization of B with the set V is a homomorphism from B to A. This gives that A 2 HCdfnmH . We need to show that IGwdfH = IGwdfHcm. The IGwdfHnm IGwdfHcm part is obvious so it is left to check that IGwdfHnm IGwdfHcm. Let A be an algebra from GwdfHnm. The unit element V of A is a normal Gws! {unit therefore it is a disjoint union of weak spaces. Denote the set of weak spaces building V by . We put U to be the union of the bases of elements of . We de ne an accessibility relation R on as: for any 1 ; 2 2 1 R2 i (9q 2 1 )(9 2 H ) q 2 2 : 31
It follows that 1 R2 ) base(1 ) = base(2 ). (Here we used that V is a normal Gws! {unit.) Let R be the transitive, symmetric, re exive closure of R. Clearly, R is an equivalence relation. We have that 1 R 2 ) base(1 ) = base(2 ). For each 2 =R we put U to be base() for some 2 , and we x a function f : U ! U that is the identity on U . Now we can map the sequences of V into subsets of ! U in a following way. For any q 2 2 2 =R
m(q) def = fs 2 ! U j f s = q; s qg: Claim 3.19 The statements below hold. For all q; q0; s 2 V (i). m(q) and m(q0 ) are disjoint if q and q0 are distinct. (ii). s 2 m(q) ) s[i=u] 2 m(q[i=f (u)]) if u 2 U . (iii). s 2 m(q) ) s 2 m(q ) if 2 H . Proof (i): Suppose that s 2 m(q) \ m(q0). Then q s q0 so q and q0 are in the same weak space of V , say in . Since 2 for some 2 =R we infer that q = f s = q0 . (ii): Suppose that s 2 m(q), q 2 2 2 =R. Since q q[i=f (u)] we have q[i=f (u)] 2 2 . Now f s[i=u] = q[i=f (u)], s[i=u] s q q[i=f (u)]
cll1
shows that (ii) holds. (iii): Suppose that s 2 m(q), q 2 2 2 =R. If 0 is the weak space of q then R0 and so R 0 , 0 2 . We have s q, f s = q and therefore f s = q and s q (we used that is bounded). We put V =
[
q2V
m(q). (ii) of the previous claim shows that V is a com-
pressed Cartesian space, (iii) gives that s (V ) = V whenever 2 H .
Claim 3.20 The statements below hold. [ m(q0 ). (i). fs 2 V j 9u 2 U s[i=u] 2 m(q)g = (ii). fs 2 V j s 2 m(q)g =
q0 2ci fqg
[ q0 2s fqg
m(q0 ). 3:19(ii)
Proof [ (i): 0 If s[i=u] 2 m(q) then s = s[i=u][i=s(0 i)] 2 m(q0 [i=f (s(i))]) m(q ). On the other hand, if s 2 m(q ) for some q 2 ci fqg then q0 2ci fqg
3:19(ii)
s[i=q(i)] 2 m(q0 [i=f (q(i))]) = m(q) since f (q(i)) = q(i); q0 [i=q(i)] = q. (ii): If s 2 V ; s 2 m(q) then s 2 m(q0 ) for some q0 2 V . By Claim 3.19 (iii), s 2 m(q ). Claim 3.19 (i) implies m(q) \ m(q0 ) 6= ; ) q0 = q. 32
cll2
So q0 2 s fqg. On the other hand, if s 2 m(q0 ) for some q0 2 s fqg then s 2 m(q0 ) = m(q). We arrived to the point when the algebra in which A is embeddable and the representation function rep can be de ned: A0 = P (V ) 2 GwdfHcm; for any a 2 A; rep(a) =
[
q2a
m(q):
Clearly, rep is a Boolean embedding of A into A0 since m(q) is not empty (q 2 m(q)), and m(q) and m(q0 ) are disjoint when q and q0 are distinct (see Claim 3.19(i)). To see that[ rep preserves we compute: [ [the extra Boolean [ operations [ m(q0 ) = ci rep(a) = ci m(q) = ci m(q) 3:20( m(q0 ) = = i) q2a
q2a
q2a q0 2ci fqg
q0 2ci a
q2a
q2a
q2a q0 2s fqg
q0 2s a
rep(ci a). [ [ [ [ [ m(q0 ) = m(q0 ) = s rep(a) = s m(q) = s m(q) 3:20( = ii) rep(s a).
Before proving Theorem 2.11 (iii) we state an important result.
Lemma 3.21 Let H be a set of quasi-bounded transformations, A be a GwdfH , 1 and 2 be weak spaces of A. If (9q 2 1 )(9 2 H ) q 2 2 , then base(1 ) base(2 ).
Proof Suppose that u 2 base(1 ), q 2 1, 2 H , j?1((k))j < ! (there is a proper k 2 ! while is quasi-bounded), q 2 2 . Since 1 is a weak space, q[(k)=u] 2 1 . But then fi 2 ! j q[(k)=u] (i) 6= q (i)g ?1 ((k)) is nite so q[(k)=u] q 2 2 . The sequence q[(k)=u] of 2 takes the value u at k so u 2 base(2 ). Proof of Theorem 2.11 (iii) We need tonmshow that under the conditions formulated in the theorem GwdfH IGwdfH . Let A be an algebra from GwdfH . As we did in the proof of (ii) we put V to be the unit element of A, to be the set of weak spaces of A, R to be the accessibility relation on , R to be the equivalence relation generated by R. We show that R0 ) base() = base(0 ) (and so R 0 ) base() = base(0 )). Suppose that R0 . Then (9q 2 )(90 2 H ) q 0 2 0 . We know that 0 has a quasi-inverse in [H ], say 1 : : : n . For any i 2 n + 1 we put i to be the weak space of q 0 : : : i . Lemma 3.21 implies that base(i ) base(i+1 ) (i 2 n). If we note that 0 1 : : :n Id, and so = n then we have base() base(0 ) base(1 ) : : : base(n ) = base(). 33
l-qb
Now we can de ne the set U to be base() for some 2 2 =R. Surely, the de nition is independent from the choice of . We put f : U ! U f g to be the natural embedding (f (u) = hu; i), and : V ! =R to be the function that associates the R class of the weak space of q to a q 2 V . We de ne for any a 2 A; rep(a) = ff (q) q j q 2 ag A0 = frep(a) j a 2 Ag 0 A = hA0 ; [; ?; s ; ck i 2H;k2! It is straightforward to verify that A0 2 GwdfHnm and rep is an isomorphism between A and A0 .
3.4 Quasi-axiomatizing ICdfH
To nd a proof to Theorem 2.15, we examine the situation described in Example 2.13. The key idea is that the presence of a constant sequence can be captured by a quasi-equation. Among those GwdfG 's that have constant sequences in their universes, the simplest in structure are the ones with unit elements of the form ! U (q) for some set U and constant sequence q. A unit element of this form will be called horizontal. (It is easy to see that s (! U (q) ) = ! U (q) .) Any algebra from GwdfH that has a constant sequence q in it's universe can be homomorphically mapped into a GwdfH with horizontal unit element. (Indeed, the relativization with ! U (q) | the function that maps the element x of the algebra into x \ ! U (q) | will be a satisfactory homomorphism.) So any GwdfH that is isomorphic to a CdfH has a homomorphism into some GwdfH with horizontal unit element. The next theorem states that this requirement is not only necessary but also sucient to force a GwdfH to be isomorphic to a H {cylindric set algebra.
Theorem 3.22 Let H be an arbitrary subset of ! ! with the property that for any 2 H , is bounded and has a right quasi-inverse in ! !. Then any algebra A 2 GwdfH , that has a homomorphism into some GwdfH with horizontal unit
th-csg
element, is isomorphic to a H {cylindric set algebra. That is,
(9 set U )(9u0 2 U )(9h : A ! P (! U (hu0 ;u0 ;:::i) )) ) A 2 ICdfH : Before the proof of the Theorem above we present an important lemma.
Lemma 3.23 Suppose U is a set, u0 2 U . There is a cardinal for that if P (W 0 ) is a full compressed H {cylindric set algebra with base U 0 of cardinality bigger than then there is a homomorphism h0 : P (! U (hu0 ;u0 ;:::i) ) ! P (W 0 ). Proof Let be a cardinal bigger than !, the cardinality of U and that of H . 34
lm-d->s
For the algebra P (W 0 ), we de ne the set of weak spaces, the accessibility relations R and R on as we de ned in the proof of Theorem 2.11 (ii). For each weak space 2 we x a sequence q in . We put U 0 ( 2 =R) to be the set fq (i) j 2 ; i 2 !g.
Claim 3.24 The cardinality of jU 0 j + jU j is less than . Proof For a given 1 2 and 2 H there is exactly one 2 2 with 1 R2 . This can veri ed by noting that from q 2 2 , q 2 1 we can infer q0 2 2 when q0 2 1 ( is bounded so q q0 ) q q0 and q 2 2 \ 02 ) 2 = 02 ). For any 2 H , is right quasi-invertible in ! ! so there is a transformation f 2 ! ! with f Id. Suppose for contradiction that there is a k 2 ! for that jf?1 (k)j = !. Then the transformation f takes the value (k) on any coordinate of the in nite set f?1 (k). But this is impossible since f takes the value i on coordinate i for all but nitely many i 2 !. This reasoning shows that f is bounded. For a given 2 2 and 2 H there is exactly one 1 2 with 1 R2 . Suppose that 1 ; 01 2 , q 2 1 , q0 2 01 and q 2 2 3 q0 . Then q q0 and q f q0 f since f is bounded. But this implies that q q f q0 f q0 , q 2 1 \ 01 so 1 = 01 . For a given 2 , jf0 2 j R0 or 0 Rgj 2 jH j. For a given 2 , we have that 0 2 and sit in the same R class if and only if there is a nite series 0 ; : : : n of weak spaces with 0 = , n = 0 and i 2 n ) (i Ri+1 or i+1 Ri ). Therefore, the size of a R class is less than 1 + 2 jH j + (2 jH j)2 + : : : ! jH j. Finally, jU 0 j ! j j ! ! jH j = ! jH j, jU 0 j + jU j < . Let f ( 2 =R) be any function mapping U 0 onto U with the property (8i 2 !)(8 2 ) f (q (i)) = u0 : Such an f exists since not more than jU 0 j-many elements of U 0 are to be mapped on u0 so there are enough elements left to cover the set U ? fu0 g. With the help of f , for each sequence q of ! U (hu0 ;u0 ;:::i) a subset of W 0 can be associated.
m(q) = fs 2 W 0 j f s = q; s 2 2 2 =Rg: As it was presented in the more[complicated situation in the proof of Theorem 2.11 (ii) the function h0 (a) = m(q) is a homomorphism. (Claims 3.19 and q2a
3.20 hold here as well since the original proofs can be adapted to the present situation. ) 35
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Proof of Theorem 3.22 In Theorem 2.11 (ii) and (iii) IGwdfH = IGwdfHcm was proved under the conditions that for any 2 H , is quasi-bounded and it has a quasi-inverse in [H ]. This holds in our situation so we may assume that A 2 GwdfHcm. The unit element of A is V ! U for some set U . Let U 0 be
a set with cardinality bigger than the cardinality of U and the cardinal that corresponds to the set U according to Lemma 3.23. Let f be a function mapping U 0 onto U . With the help of f , for each sequence q of ! U a subset of ! U 0 can be associated. Let V 0 =
[
m(q) = fs 2 ! U 0 j f s = qg:
m(q). As it was presented in the proof of Theorem 2.11 (ii) the [ function r1 : A ! P (V 0 ); r1 (a) = m(q) is an embedding. q2V
q2a
Let W 0 = ! U 0 n V 0 . Just like V 0 , W 0 is a Gws! {unit and s (W 0 ) = W 0 for any 2 H . So W 0 can be a unit element of a GwdfHcm . According to Lemma 3.23 there is a homomorphism h0 mapping P (! U (hu0 ;u0 ;:::i) ) into P (W 0 ). Let r2 be the composition of the homomorphisms h0 and h (r2 = h0 h) mapping A into P (W 0 ). Since P (! U 0 ) = P (V 0 ) P (W 0 ) the function rep de ned as rep(a) = r1 (a) [ r2 (a) is an embedding and it demonstrates that A 2 ICdfH . In the second part of this subsection we show that if an algebra in GwdfH satis es Qx then it has a homomorphism to GwdfH with horizontal unit element. We assume that H satis es the assumption stated in Theorem 2.19.
Lemma 3.25 If A 2 GwdfH and A j= Qx then there is an ultra lter U of A with the property that ?c(?) x 2 U holds whenever x \ s x = 0 for some ? ! ! 2 [H ] with d? = Idd?. ^
Claim 3.26 8; 1; : : : ; n 2 [H ] 90 ; 10 ; : : : ; n0 2 [H ] : 0 i = i0 . in Furthermore, if d? = 1 d? = = n d? = Idd? then the same holds to 0 ; 10 ; : : : ; n0 (? ! !). Proof We prove by induction. If n = 0 then 0 = . Suppose that the statement holds for n ? 1. Let 00 ; 100 ; : : : ; n00?1 be associated to ; 1 ; : : : ; n?1 . Let and n be the left quasi-inverse of 00 n and respectively. Since [H ] contains all nite 9 transformations we can assume that 00 n (k) = 8 = < k 2? 0 0 00 0 k 2 m ? ? 0 ; = n (k) for some m0 2 !. Setting = , : k m0 k i0 = i00 (i = 1; : : : ; n) gives the required result. 36
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Proof of Lemma \3.25 Let B = f?c(?)x : ? ! !; 9K ! [H ] ( 2 K ! d? = Idd?); s x = 0g. Qx states that B has the nite intersection 2K property.T We put B to be B 's lter closure, that is, B = fy 2 A : 9X ! B y X g. Claim 3.27 B is closed under s for all 2 [H ]. \ Proof Suppose ?c(?)x 2 B, s x = 0 for some ? ! !; K ! [H ], 2K 2 K ! d? = Idd?. Let ?0 = fi 2 ! ? ? : (i) 2 ?; 8j 2 i ? ? ( (i) = 6 (j ))g. 0 0 is bounded so ? is nite. If ? = f 1 ; : : : ; n g then let 0 = [ 1 ; ( 1 )] : : : [ n ; ( n )]. Let ?00 = ?0 [ (? ?f ( ) : 2 ?0 g), 1 = (0 )[?=Id]. Now we have that 0 1 d! ? ? = d! ? ? and for all i 2 ? 1?1 (i) = fig. Using Claim 3.26, 0 0 we pick\corresponding and ( 2 K ). Then 0 = \ transformations \ 1 and to \ s0 s10 s x = s0 s10 s x = s0 s 0 s1 x = s0 00 s0 s1 x. Not 2K
2K
2K
2K
ing that 0 0 0d?00 = Idd?00 we have ?c(?00 ) s0 s1 x 2 B . So ?c(?00 ) s0 s1 x =6 ?s0 c(?) s1 x =6 ?s0 s1 c(?) x =5 ?s c(?) x = s (?c(?) x) gives s (?c(?) x) 2 B . Finally, suppose that x 2 B . Then there are x1 ; : : : ; xn 2 B with x n \
n \
n \ 1
xi .
So s x s xi = s xi 2 B giving s x 2 B . 1 1 Let U be a maximal lter extending B with the property that it is closed under s for all 2 [H ]. We can use Zorn's lemma to construct such a lter. We show that U is indeed an ultra lter. Suppose not. Then there is a y such that neither y nor ?y\is in U . y cannot be added to \ U so there are \ z 2 U , K ! [H ] such that z \ s y = 0. Then 0 = z \ s y s (z \ y) giving 2K
2K
2K
?(y \ z ) 2 B . Similarly, ?(?y \ z 0 ) 2 B . Therefore U 3 ?(?y \ z 0 ) \?(y \ z ) = ?((y [ z 0 ) \ (z [ z 0) \ (z [ ?y)). But all the three terms in the last expression are in U so their intersection is in U giving that U contains an element and its complement.
Our algebra A 2 GwdfH is a normal Boolean algebra with operators. (Cf. e.g. [HMTI] De nition 2.7.1.) We can de ne its canonical embedding algebra or ultra lter extension A as follows (see De nition 2.7.4. in [HMTI]). Let V be the set of ultra lters of A. If O is any of the operators fs : 2 H g[fci : i 2 !g and X V then we de ne OX = fU 2 V : (9S 2 X )(8x 2 U ) Ox 2 S g: Then A is hP (V ); [; ?; s ; ci i 2H;i2! . 37
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As stated in Corollary 3.5, IGwdfHnm is axiomatized by (*1){(*7) (de ned in De nition 3.3) together with the de nition of ci (i 2 !). All that axioms but ci (?ci x) = ?ci x and s (?x) = ?s x are positive, that is, they do not contain any occurance of the symbol ? for complementation. But the two exceptions are equivalent (in the presence of the others) with some positive formulas:
ci (?cix) = ?ci x i ci 0 = 0; ci (x ^ ci y) = cix ^ ciy s (?x) = ?s x i s (x ^ y) = s x ^ s y: Since positive equations valid in a Boolean algebra with operators still hold in the ultra lter extension of the algebra, (see e.g. [HMTI] Remark 2.7.15) we can conclude that the ultra lter extension A of A 2 GwdfHnm is in GwdfHnm. Furthermore, under the hypothesis of Theorem 2.19, IGwdfH = IGwdfHnm holds, so the conclusion stated above holds to the class GwdfH as well. Finally, we note that every Boolean algebra with operators is embeddable into its ultra lter extension. (The map that associates fU 2 V : x 2 U g to an element x in A will suce.)
Lemma 3.28 If A 2 GwdfH and A is its ultra lter extension then the atom a = fU g has the property x \ c(?) a s x (? ! !; 2 [H ]; d? = Idd?; x 2 A ) provided that U is an ultra lter of A that satis es the property stated in Lemma
ll2
3.25.
Proof It is enough to show that the condition x \ c(?)a s x holds for all atoms of A . Suppose x = fF g. If F 62 c(?) a then there is nothing to show. If F 2 c(?) a then 8y 2 F c(?) y 2 U . Suppose for contradiction that x 6 s x, that is, 9y 2 F s y 62 F . Then ?s y 2 F , w = y \ ?s y 2 F . We can conclude that c(?) w 2 U . On the other hand, w \ s w = 0 gives ?c(?)w 2 U and that is a contradiction.
Lemma 3.29 If A 2 GwdfHcm and A has an atom a with x \ c(?)a s x whenever ? ! !; 2 [H ]; d? = Idd?; x 2 A then A has a homomorphism
ll3
to a GwdfH with horizontal unit element.
Proof We denote the base of A by U 0 and the [ universe of A by A . Let q be a sequence in a, V = fq : 2 [H ]g, V = c(m)V . Put U0 to be the m2! range of q. Fix an arbitrary element u 2 U0 . Denote U 0 ? U0 [ fug by U . Let f be the function that is the identity on U 0 ? U0 and the constant u on U0 . Claim 3.30 x 2 A; s; s0 2 V ; s 2 x; f s = f s0 ! s0 2 x. Proof Let ? = fi 2 ! : s(i) 62 U0g. Since s q for some 2 [H ], ? is nite. Clearly, ? = fi 2 ! : s0 (i) 62 U0 g so sd? = s0 d?. We have that s0 q 0 for some 38
clll3
0 2 [H ]. s = q [?=s], s0 = q 0 [?=s]. Let and 0 be the right quasi-inverses of and 0 . Since all the nite transformations are in [H ], we can choose and 0 that they satisfy the following additional properties: d? = 0 d? = Idd?, d! ? ? = 0 0 d! ? ?. Then s = (q [?=s]) = q [?=s] = q 0 0 [?=s] = (q 0 [?=s]) 0 = s0 0 . Since s 2 x \ c(?) a s x, we get that s0 0 = s 2 x. (We used that a s a and so q 2 a.) If we suppose for contradiction that s0 62 x then s0 2 ?x \ c(?) a s 0 ? x and so s0 0 2 ?x but that cannot be the case.
Now we can de ne a homomorphism that maps A into P (! U (hu;u;:::i) ): h(x) = ff s : s 2 x \ V g: It is straightforward to verify that h preserves \, while Claim 3.30 proves that it preserves complementation. The following two claims prove that h preserves the extra Boolean operations. Claim 3.31 cih(x) = h(ci x). Proof r 2 ci h(x) ) r = (f s)[i=r(i)] for some s 2 x. But r = (f s)[i=r(i)] = f (s[i=r(i)]) 2 h(ci x) since s[i=r(i)] 2 ci x. On the other hand, r 2 h(ci x) ) r = f s for some s 2 ci x. So there is s0 2 x with s0 [i=s(i)] = s. But r = f s = f (s0 [i=s(i)]) = (f s0 )[i=f (s(i))]) 2 ci h(x)
Claim 3.32 s h(x) = h(s x). Proof r 2 s h(x) , r 2 h(x) , 9s0 2 x f s0 = r . r = f s for some s 2 V so f s0 = r = f s . Using Claim 3.30 we have that s0 2 x , s 2 x. But s 2 x , s 2 s x , r = f s 2 h(s x) completes the proof.
Summarizing the results of this subsection, we prove Theorem 2.19.
Proof of Theorem 2.19 Suppose that the algebra A is a model of Qx [ Eq(IGwdfH ). Lemmas 3.25 and 3.28 state that the ultra lter extension A of A has an atom with certain properties. Since the ultra lter extension satis es all the positive equations valid in the algebra, we have that A j= Eq(IGwdfH ). Using Theorem 2.11 we conclude that A is isomorphic to an algebra B 2 GwdfHcm. Lemma 3.29 implies that B has a homomorphism to a GwdfH with horizontal unit element. Since A is embeddable into its ultra lter extension and, in that way, into B, A also has a homomorphism to a GwdfH with horizontal unit element. Finally, Theorem 3.22 states that the above mentioned result is sucient to show that A 2 ICdfH . 39
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3.5 Theorem 2.30: the cylindric reduct of Mod( 1 2) H
;H
The proof will have the following structure. Any model of H1 is actually an extension of a model of EqGwdfHnm. But Theorem 2.11 (i) imply that a model of EqGwdfHnm is isomorphic to a GwdfHnm We will examine the extra abstract constant d01 . (In the presence of the substitutions the other dij 's are term de nable so it is enough to concentrate on d01 .) Finally, the proper behaviour of d01 will lead us to a representation function that proves our theorem. Recall from Notation 2.8 that if q1 are q2 sequences then q1 q2 indicates that they agree on all but nitely many coordinates.
Notation 3.33 Let H be a set of transformations of !. Suppose that = h1 : : : n i 2 H . We introduce the notation s x for the term s1 : : : sn x. Remark 3.34 Let H be a set of transformations of ! with the property that [H ] is rich. (Recall that [H ] is the subsemigroup of h! !; i generated by H [f[i=j ] j i; j 2 !g.)
nt-stau
rm-H4
In the de nition of a rich semigroup the existence of two special elements s^ and p^ were stated. We x two elements of [H ] with the required property and denote them by s^ and p^. We also x ss^ and sp^ to stand for the terms s 1 and s 2 respectively where 1 = s^, 2 = p^, and 1 ; 2 2 H . It is provable from H1 that for any choice of 1 2 H with 1 = s^, s 1 is the same operation of the algebra so ss^ (and similarly sp^) is well de ned. s^(i) 6= s^(j ) if i 6= j since p^ s^ = Id. There is a permutation f of ! that takes s^(0) to 0 and s^(1) to 1 and xes all the elements of ! ? f0; 1; s^(0); s^(1)g. (f is one of [0; s^(0)] [1; s^(1)], [1; s^(1)] [0; s^(0)] or [0; 1] depending on whether any of s^(0) = 0, s^(0) = 1, s^(1) = 0 or s^(1) = 1 hold.) Since f is nite f 2 [H ]. So there is a H 2 H for that H = f . We formalize a set of equations that are all consequences of H1,H2.
s[0=1]x s[1=0]x ci d01 s[0;1]d01 s H ss^d01
c0(d01 ^ x); c1(d01 ^ x); d01 whenever i 2; (H4) d01; d01: Here s[i;j] x stands for the term s x where 2 H and = [i; j ]. Since [H ] is = = = = =
rich, all the nite transformations (including [i; j ]) belong to [H ].
Lemma 3.35 Let H be a set of transformations of ! with the property that [H ] is rich. Suppose that A is a GwdfH extended with the abstract18 constant d01
18 Here the adjective \abstract" means that this constant is NOT necessarily the set theoretically de nable diagonal element, but any constant that satis es the required equations. It will be shown that this \abstract" constant need to have a nice structure similar to that of the set theoretic diagonal element.
40
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that satis es H4. If x is an element of A and hu0 ; u1 ; u2 ; : : :i is a sequence from the universe of A then hu0 ; u1 ; u2 ; : : :i 2 d01 ^x implies hu0 ; u0; u2 ; : : :i 2 x;
hu1 ; u0; u2 ; : : :i 2 x; hu1 ; u1; u2 ; : : :i 2 x:
Proof Suppose that hu0; u1; u2; : : :i satis es the premise of the lemma. Then hu0 ; u1 ; u2; : : :i 2 c0 (d01 ^ x) H=4 s[0=1] x. This implies hu0 ; u1 ; u2 ; : : :i [0=1] = hu1 ; u1 ; u2; : : :i 2 x. Using s[1=0] x H=4 c1 (d01 ^ x) we can get hu0 ; u0 ; u2; : : :i 2 x. s[0;1]d01 H=4 d01 implies hu1; u0; u2; : : :i 2 d01 . If we suppose for contradiction that hu1 ; u0; u2 ; : : :i 62 x then hu1 ; u0 ; u2 ; : : :i 2 ?x and according to the previous observation we can conclude that hu1 ; u1 ; u2; : : :i 2 ?x and that is impossible. Lemma 3.36 Let H; A and d01 be as in Lemma 3.35. If u and v are the rst two elements of a sequence q of d01 then any sequence r of an arbitrary x 2 A can be modi ed in any of its positions from u to v provided that q and r are in the same weak space. In other words, if q 2 d01 , r 2 x for some x 2 A, q r, r(k) = q(0) = u for some k 2 ! and q(1) = v then r[k=v] 2 x. Proof First suppose that k =6 1. Let q = (r [0; k])[1=v]. Clearly, q is in the weak space of q (and of r). Therefore, q 2 ci1 : : : cin d01 where ? = fi1; : : : in g is the set where q and q do not agree (it is nite). Now q (0) = (r [0; k])(0) = r(k) = u and q (1) = v so 0; 1 62 ? and ci d01 H=4 d01 whenever i 2. We can conclude that q 2 d01 . We put z to be an element of ! ? Rng(^s). [H ] is rich so ! ? Rng(^s) is not empty. (q p^)[z=r(1)] s^ = q 2 d01 implies that (q p^)[z=r(1)] 2 ss^d01 . (q p^)[z=v][^s(1)=r(1)] s^ = r [0; k] 2 s[0;k] x, (q p^)[z=v][^s(1)=r(1)] 2 ss^s[0;k] x. Furthermore, (q p^)[z=r(1)] = (q p^)[z=v][^s(1)=r(1)] [z; s^(1)]. We put r def = (q p^)[z=r(1)] H?1 . Summarizing all the results we get that r 2 s H ss^d01 H=4 d01 and r 2 s H s[z;s^(1)]ss^s[0;k]x, r (0) = q(0) = u, r (1) = q(1) = v. Using Lemma 3.35 we conclude that r [0=v] 2 s H s[z;s^(1)] ss^s[0;k] x. This means that r [0=v] H [z; s^(1)] s^ [0; k] 2 x. Then r [0=v] H [z; s^(1)] s^ [0; k] = ((q p^)[z=r(1)] H?1 H )[^s(0)=v] [z; s^(1)] s^ [0; k] = (q p^ [z; s^(1)])[^s(1)=r(1)][^s(0)=v] s^ [0; k] = (q p^ s^)[1=r(1)][0=v] [0; k] = ((r [0; k])[1=v][1=r(1)] [0; k])[k=v] = (r [0; k] [0; k])[k=v] = r[k=v] 2 x. If k = 1 then r 2 x ) r [1; 2] 2 s[1;2] x ) (r [1; 2])[2=v] 2 s[1;2] x ) (r [1; 2])[2=v] [1; 2] = r[1=v] 2 x. Proof of Theorem 2.27 (1) Surely, any representable cylindric set algebra is a subalgebra of a reduct of a model of H1,H2. 41
lm-3
To see the other direction, suppose that A0 is a model of H1,H2. A0 j= H 1 means that all the valid equations of GwdfHnm hold in A0 . Naturally, some operations of tH are term functions in GwsH . Since IGwdfHnm is a variety, the tH {reduct of A0 is isomorphic to a GwdfHnm , say to A. We will use only that A 2 IGwdfH . The image of d01 under the isomorphism is an element of A not necessary the set theoretic (0,1){diagonal. The image of any other diagonal constant can be computed from the image of d01 since dij is term de nable from d01 in GwdfH . This shows that to represent A0 , it is enough to represent A that is to nd a isomorphism rep mapping the cylindric reduct of A into a RCA! . The universe V of A is a Gws! {unit therefore it is a (disjoint) union of weak spaces. Denote the set of weak spaces building A by . For any weak space from , base() is the base set of . d01 de nes an equivalence relation on base() in a following way: for any u; u0 2 base()
u u0 i (9q 2 d01 \ ) q(0) = u; q(1) = u0
Claim 3.37 is re exive. Proof Suppose u 2 base(). Let hu0; u1; u2; : : :i 2 arbitrary. It is obvious that hu; u; u2; : : :i 2 . Now hu; u; u2; : : :i 2 V = c1 d01 . So (9v 2 base()) hu; v; u2 ; : : :i 2 d01 . Applying Lemma 3.35 to the case when x = d01 it gives that hu; u; u2; : : :i 2 d01 Claim 3.38 is symmetric. Proof Follows from Lemma 3.35 when applied to the case x = d01. Claim 3.39 is transitive. Proof Follows from Lemma 3.36 when applied to the case x = d01, k = 0. Our aim is to embed the cylindric reduct of the not fully represented algebra
A into a Gws! . For saving energy and not losing what we have already achieved,
we are looking for a representation function rep that acts not only on the algebra level but deeper, on the sequence level hiding in the elements of A. In other words, rep(x) = ff (q) j q 2 xg where f if the sequence level function mapping V into the universe V of some algebra A . To ensure that f (d01 ) is the set theoretic constant we want for any q; q0 2 2 that (8i 2 !) q(i) q0 (i) should imply f (q) = f (q0 ). This observation gives us a hint for an adequate de nition of f . 42
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cl-3
For every 2 , we de ne U to be the set (base()= ) fg. U corresponds to the classes of base(). Clearly, if and 0 are distinct then U and U0 are disjoint. For every q 2 , we de ne
f (q) = hhq(0)= ; i; hq(1)= ; i; hq(2)= ; i; : : :i The f image of a 2 is also a weak space in ! U . Let it be denoted by . Now we can de ne the image algebra A 2 Gws! :
V =
[ ;
2
A = frep(x) j x 2 Ag; A = hA ; \; ?; ci ; dij i: One can easily verify that the above de ned rep is an isomorphism between the cylindric reduct of A and a generalised weak cylindric set algebra. Finally, the result IGws! IGs! = RCA! published in [HMTII] Theorem 3.1.103. completes the proof.
3.6 Further observations on Mod( 1 2) H
;H
We have seen that a model A0 of H1,H2 is isomorphic to an algebra A whose tH reduct is a GwdfH . In the previous proof another isomorphism rep was de ned that mapped the cylindric reduct of A into A 2 Gws! . A can be expanded with the abstract operations s ( 2 H ) as follows
s y def = rep s rep?1 y: The expanded algebra is denoted by A as well.
We examine the behaviour of the new, abstract operations. In this subsection we collect plenty of the characteristic features that will be useful in the proof of Theorem 2.30 (3). All the notations (A; A ; ; ; f; rep) that were introduced in the previous subsection are kept. The set of weak spaces of A is denoted by . and is in a one-to-one correspondence in a natural way. We have that base( ) = (base()= ) fg. >From now we suppose that all the elements of H are bounded transformations of !.
Claim 3.40 If 2 H (and so is bounded) and 1 ; 2 2 then the statements below hold.
s 1 2 s 1 2
or i
s 1 \ 2 = ;. (9q 2 2 )(q 2 1 ). 43
cl-bound
(82 2 )(9!1 2 )(s 1 2 ). (Here 9! stands for `there exists exactly one'.)
Proof Suppose that q 2 s 1 \ 2 . For all q0 2 2, q0 2 1 since q 2 1 and fi 2 ! j q (i) 6= q0 (i)g ?1 fi 2 ! j q(i) 6= q0 (i)g is nite. So 2 s 1 . To prove the last statement suppose that q 2 2 2 . Put 1 to be the weak space of q . Clearly, q 2 1 and so 2 s 1 . If 0 2 and 2 s 0 then s 1 \ s 0 = s (1 \ 0 ) 2 shows that 1 \ 0 6= 0 and so 1 = 0 . That claim shows that the following de nition is explicit.
De nition 3.41 For each 2 H , we de ne a transformation ^ of : ^ (2 ) = 1 if 2 s 1 :
d-sonL
Since and are in one-to-one correspondence, ^ can be viewed as a transformation of : 1 ; 2 2 , ^ (2 ) = 1 if ^ (2 ) = 1 . We present an important property of these transformations.
Claim 3.42 Suppose that 1 ; : : : ; n and 10 ; : : : ; m0 are elements of H . If n : : : 1 m0 : : : 10 then ^1 : : : ^n and ^10 : : : ^m0 are the same
cl-hatjo
transformations of (and of as well).
Proof Suppose that 2 . ^1 : : : ^n () = 0 if sn : : : s1 0. Then for any q 2 , q n : : : 1 2 0 . Since n : : : 1 m0 : : : 10 and they are bounded, q n : : : 1 q m0 : : : 10 . Then q m0 : : : 10 2 0 , sm0 : : : s10 0 , ^10 : : : ^m0 () = 0 and this was to prove. Remark 3.43 The domain of the function^de ned in De nition 3.41 is H . The previous claim implies that^can be de ned on [H ] as well. If 2 [H ] ! ! then we set ^ def = ^1 : : : ^n where n : : : 1 . Clearly, ^ does not depend on the choice of 1 ; : : : ; n . The new^: [H ] ! has the following properties: If ; 2 [H ]; then ^ = ^. If ; 2 [H ] then d = ^ ^. Claim 3.44 When ^(2 ) = 1 for some 2 H; 1 ; 2 2 then base(2 ) base(1 ), 2 = 1 \ 2 base(2 ). 44
r-ph
cl-base
Proof Suppose that 2 s 1 , q 2 2. For all u 2 base(2 ), q[(0)=u] 2 2 and so q[(0)=u] 2 1 . But (q[(0)=u] )(0) = u 2 base(1 ). It shows that base(2 ) base(1 ). To prove the second statement suppose that u; u0 2 base(2 ), u 2 u0 . There is a q2 2 d01 \ 2 for that q2 (0) = u, q2 (1) = u0 . We x a k 2 ! for that (0) = 6 (k) and a nite transformation h of ! that maps (0) to 0 and (k) to 1. Since [H ] is rich h 2 [H ] and so there is 2 H for that = h. We put = q2 h [1; k]. q2 2 d01 and s s s[1;k] d01 = dh(([1;k](0)));h(([1;k](1))) = q1 def d01 implies that q1 2 d01. q2 2 2 and 2 s 1 gives that q1 2 1. Finally, q1 (0) = u, q1 (1) = u0 shows that u 1 u0 . For the other direction suppose that u; u0 2 base(2 ), u 1 u0 . There is a q 2 1 \ d01 for that q(0) = u, q(1) = u0. Pick any q2 2 2 with q2 (0) = u, = q2 h [1; k]. As in the previous case q1 2 1 . q2 (1) = u0 . We put q1 def q q1 so they disagree on a nite set ? = fi1 ; : : : ; in g. 0; 1 62 ?. We conclude that q1 2 ci1 : : : cin d01 = d01 . Using again s s s[1;k] d01 = d01 and q1 2 d01 we get that q2 2 d01 . This proves that u 2 u0 . De nition 3.45 We de ne the function [ [ b: fh; ig base( ) ! base( ) 2 H; 2
d-em
2
as follows. b; (h ; i) = h 0 ; ^ ()i if 0 2 base(^())=^ () and 0 holds. The previous claim proved that b is well de ned. We collect some properties of the function b de ned above. Claim 3.46 (1). q 2 2 ) b; q 2 V . (2). q; q0 2 2 ) b; q b; q0 . (3). b; is injective. (4). (8 2 )(9!0 2 )(8q 2 ) b; q 2 0 . Proof (1): Since q 2 V , there is a s 2 V for that f (s) = q. The weak space of s in A is , and that of q in A is . We show that b; q = f (s ) 2 V by checking that they take the same value on an arbitrary i 2 !. (b; q )(i) = b; (q((i))) = b; (f (s)((i))) = b; (hs((i))= ; i) = hs((i))=^ () ; ^ ()i = f (s )(i) To prove (2) we need only that is bounded. (3) follows from the de nition, and (4) from (2).
45
cl-propb
Remark 3.47 If a function g satis es condition (4) of Claim 3.46 then for each 2 H , a transformation ^g of can be de ned: ^g () = 0 i (8q 2 ) g; q 2 0 : When g coincides with b then we get back ^ , that is ^b = ^ . Claim 3.48 Let 2 , 1 ; : : : ; n and 10 ; : : : ; m0 be elements of H . (5). If ^1 : : :^n ( ) = then b1 ;^2 (:::^n ( ):::) b2 ;^3 (:::^n ( ):::) : : :bn; is the identity mapping of base( ). (6). If ^1 : : : ^n ( ) = ^10 : : : ^m0 ( ) then b1 ;^2 (:::^n ( ):::) b2 ;^3 (:::^n ( ):::) : : : bn ; = = b10 ;^20 (:::^m0 ( ):::) b20 ;^30 (:::^m ( ):::) : : : bm0 ; Proof First we note that the expression formulated above is correct. The range of bi ;^i+1 (:::^n ():::) is contained in base(^i (: : : ^n () : : :)) and that is the domain of bi+1 ;^i (:::^n ():::) . To check the validity of the expression in (6) we evaluate the two sides of the equation on an arbitrary element of base( ), say an hu= ; i. When the left side of the equation is applied to that element we get hu=(^1 :::^n )() ; (^1 : : : ^n )()i and that is the same point as the image of hu= ; i under the function that appear on the right side of the equation. (5) follows from the de nition.
Notation 3.49 We de ne a new composition operation mapping V H into V : q def = b;[q] q where [q] is the weak space of q in A . Now a direct de nition can be given to s ( 2 H ). Claim 3.50 For all 2 H; y 2 A s y = fq 2 V j q 2 yg Proof As a side result, in the proof of Claim 3.46 (1) we showed that if s 2 V; q 2 V ; f (s) = q then f (s ) = q . Now we can compute that q 2 s y , s 2 s rep?1 y , s 2 rep?1 y , f (s ) 2 y , q 2 y. We arrived to the point when a new class of set algebras can be introduced. A new algebra has a deeper structure than a GwsH does and its operations are computable from the structure of the algebra. 46
rm-sigmab
cl-prop5
nt-circ
cl-newd2
De nition 3.51 An algebra A = hA; [; ?; ck ; dkl ; s ik;l2!;2H is called a nonstandard generalised weak H {cylindric set algebra with equality (or brie y a nstGwsH ) if hA; [; ?; ck ; dkl ik;l2! is a generalised weak cylindric set algebra
df-nonst
and there is a function b with the following properties: Let V be the unit element of A. We[put to be the set of weak spaces building V . The domain of b is f; g base(), the range is a subset of
[
2
base().
2H;2
b satis es the properties (1)-(6) listed in Claim 3.46 and Claim 3.48 where
^ = ^b is de ned as in Remark 3.47. We put : V H ! V to be the function q def = b;[q] q . Clearly, the properties of b gives that b;[q] q is an element of V . Then s y = fq 2 V j q 2 yg. If A is a nstGwsH and b is a function with the required properties then b is
called a base transformation function of A. Note that the base transformation function is not uniquely determined. The theorem below is a summary of the results that have achieved so far in this subsection.
Theorem 3.52 Any model of H1,H2 is isomorphic to a nstGwsH . Theorem 3.53 GwsH nstGwsH . Proof Any GwsH is a nstGwsH with the identity as base transformation function. (8 2 H )( 2 )(8u 2 base()) b;(u) def = u. Claim 3.54 Suppose that A 2 nstGwsH . Then Claim 3.42 is true in A. Proof As usual, is the set of weak spaces of A. Pick an arbitrary element of , say , and an arbitrary sequence q in . Using the de nition of ^ in Remark 3.47 we have that ^1 : : : ^n () and ^10 : : : ^m0 () are the weak spaces of b1 ;^2 (:::^n ():::) b2;^3 (:::^n ():::) : : : bn; q n : : : 1 = q n : : : 1 and b10 ;^20 (:::^m0 ():::) b20 ;^30 (:::^m ():::) : : : bm0 ; q m0 : : : 10 = q m0 : : : 10 each respectively. Since n : : : 1 m0 : : : 10 , we have that q n : : : 1 q m0 : : :10 . This fact and Claim 3.48 implies that q n : : :1 q m0 : : :10 and so they are in the same weak space, that is ^1 : : : ^n () = ^10 : : : ^m0 (). 47
th-nonst th-st-nst
cl-hatjo2
Remark 3.55 As we did in Remark 3.43, we can introduce the function ^ : [H ] ! as ^ def = ^1 : : : ^n when 2 [H ]; 1 ; : : : ; n 2 H and n : : : 1 .
r-pb
In the same sense, the domain of b can be expanded
b:
[
2[H ];2
f; g base() !
[
2
base()
= b1 ;^2 (:::^n ():::) b2 ;^3 (:::^n ():::) : : : bn ; b; def where 2 [H ]; 1 ; : : : ; n 2 H; 2 and n : : : 1 . If 10 ; : : : ; m0 2 H and m0 : : : 10 then ^1 : : : ^n = ^ = ^10 : : : ^m0 so ^1 : : : ^n () = ^10 : : : ^m0 (). This gives that (6) of Claim 3.48 is applicable showing that the above de nition of b does not depend on the choice of 1 ; : : : ; n . We collect some properties of our new^functions. ; 2 [H ]; 2 . (1). implies ^ = ^. (2). d = ^ ^. (3). Id implies that b; is the identity mapping of base(). (4). b ; = b;^() b; . (5). ^() = ^() implies b; = b; . (1) and (2) can be found in Remark 3.43, (3) and (5) are only reformulations of (5) and (6) of Claim 3.48, and (4) is a consequence of (6) of Claim 3.48.
Lemma 3.56 Let A be a subdirect irreducible model of H1,H2,H3. Then either ?c0 (?d01 ) = 1 or ?c0 (?d01 ) = 0. Proof Suppose that in a model A of H1,H2,H3 neither ?c0(?d01) = 0 nor c0 (?d01) = 0 holds. We show that the relativizations with ?c0(?d01) and c0 (?d01) are homomorphisms, but not isomorphisms, and A is a subdirect product of the two image algebras. This proves that A cannot be subdirectly
th-subd
irreducible. A relativization with some element R is a homomorphism if and only if R is closed under the operations of the algebra. H3 implies that ?c0 (?d01 ) and c0 (?d01) are closed. In the image algebras ?c0(?d01) = 1 or c0 (?d01) = 1 will hold showing that they cannot be isomorphic to A. Finally, an element of A is solely determined by its two image under the two relativizations.
Lemma 3.57 If ?c0(?d01) = 1 holds in A 2 nstGwsH then A is a GsH . 48
l-extr
Proof Let A be the universe, V the unit element, the set of weak spaces of A. ?c0 (?d01 ) = 1 shows that for all 2 , jbase()j = 1. Introduce the set theoretical operations on A:
~s x def = fq 2 V j q 2 xg: Then A0 def = hA; [ ?; ck ; dkl ; ~s ik;l2!;2H 2 GsH . We show that ~s = s and so A = A0 2 GsH . s x = x since the axiom s (x ? c0(?d01 )) = x ? c0(?d01) in H3 can be reduced to that form using ?c0 (?d01 ) = 1. On the other hand, ~s x = x is a result of that fact that in V all the sequences are of the form hu; u; u; : : :i for some u.
3.7 The witness tree method
In this subsection we build a machinery called the witness tree method to prove Theorem 2.30. The essence of the method is the following. The failure of an equation in a given algebra can be captured by a tree structure that re ects the local nature of the algebra. This tree can be cut out from the algebra and moved to another one with better properties to testify the failure of the same equation therein. In our case we will show that an equation that is not satis ed in a nstGwsH cannot hold in all the GwsH s.
De nition 3.58 Let H be a set of transformations of !. We say that = hV; E; r; U; ; "; li is a witness tree of type H if the following requirements are
ful lled. (i). hV; E i is a directed tree. r 2 V . The edges are directed from the root r towards the leaves. (ii). U; and " are functions with domain V . If v 2 V then Uv is a set, v is a term of type t=H , and "v is either 3 or 63. (Note that we used the less conventional notation Uv instead of U (v).)
(iii). l is a function with domain E . If e = vv~ 0 2 E is an edge then le is one of '=', '[k=u]' or '' where k 2 !, u 2 Uv , 2 H . We will call le the label of the edge e. (iv). If e = vv~ 0 2 E is an edge then Uv Uv0 . Furthermore, if le is not '' for some 2 H then Uv = Uv0 . (v). For every vertex v, v , "v , the number of children of v, the labels of the edges pointing to the children of v, and the value of and " on of the children vertexes of v are in a connection indicated in the following table. 49
d-wt
v , " v (?%); 3 (?%); 63 (%1 [ %2 ); 3 (%1 [ %2 ); 63
ck %; 3 ck %; 63 s %; 3 s %; 63 X; 3 X; 63 dkl ; 3 dkl ; 63
The number of children 1 1 1 2 1
jUv j 1 1 0 0 0 0
The label of The value an edge of , " on pointing to a a child child vertex = %; 36 = %; 3 = %i ; 3 = %i ; 63 [k=u] %; 3 [k=u] %; 63
? ? ? ?
%; 3 %; 63
where i is 1 or 2 for i 2 f1; 2g where u 2 Uv for each u 2 Uv
? ? ? ?
De nition 3.59 Let be a witness tree. We de ne a function : V ! ! ! by recursion. For the root r, r = Id. If e = v1~v2 2 E and v1 is already de ned then v2 is set to v1 when le is '=' or '[k=u]' for some k 2 !; u 2 Uv1 and to v1 when le is ''. De nition 3.60 Let % be a term. Then M% denotes the set of those transformations f of ! for that exist 1 ; : : : ; n 2 H with the property that f = 1 : : : n 6 j ) but and s1 ; : : : ; sn are letters of %. We allow i to be equal with j (i = in that case si and sj refer to the multiple occurrence of the same letter in %. Clearly, n is smaller than or equal to the length of term %. Claim 3.61 The set M% de ned in the previous de nition is nite. Proof The term % has only nite number of letters so there is only nitely many possible choices of 1 ; : : : ; n .
De nition 3.62 Let be a witness tree. Then M denotes M r . Claim 3.63 For each vertex v of , v 2 M . De nition 3.64 M =; M 6= denotes the following subsets of M M . M = def = fh; 0 i j ^ (9 vertex v; v0 )( = v ^ 0 = v0 )g M 6= def = fh; 0 i j 6 0 ^ (9 vertex v; v0 )( = v ^ 0 = v0 )g 50
d-nuv
d-Mr
cl-Mfin
d-M cl-nuM d-M=
Both M = and M 6= are nite.
De nition 3.65 Let be a witness tree. We say the is not degenerated if (z) (8 vertex v1 ; v2 )(8 2 [H ]) v1 v2 ) Uv1 Uv2
d-degen
holds.
De nition 3.66 Let be a witness tree. We say that a function Q is a valu-
d-validq
Claim 3.67 A valuation Q of the vertex set of a witness tree and the value
cl-vq
ation of the vertex set of if the conditions below hold. (1). The domain of Q is the vertex set of . (2). For all vertex v, Qv is an !-sequence over Uv . (3). If v1 and v2 are two adjacent vertexes of then the connection between Qv1 and Qv2 is determined by the label of the edge that joins v1 and v2 . If the label is '=' then Qv2 = Qv1 . If the label is '[k=u]' then Qv2 = Qv1 [k=u]. If the label is '' then Qv2 = Qv1 .
of Q on the root of uniquely determine each other.
Proof If we know the value of Q on the root then we can compute the values of Q on other vertexes going step-by-step towards the leaves using (3) of De nition 3.66. To verify (2), we need only that Uv1 Uv2 whenever v1~v2 is an edge. (See the table in De nition 3.58.)
Claim 3.68 Let be a witness tree with root r, Q be a valuation of the vertex set. Then
(9c1 2 !)(8 vertex v)(8i 2 !) Qv (i) 6= (Qr v )(i) ) i c1 :
Proof To get an insight into the relation of Qv and Qr v , we take an example rst. Suppose that we nd the sequence 1 , [k1 =u1], 2 , [k2 =u2], 3 , 4 , [k3 =u3], 5 of labels on the r ! v path. Then 8 if i 2 5?1 (k3 ); > > < uu32 if i 2 (3 4 5 )?1 (k2 ) ? 5?1 (k3 ); Qv (i) = > u if i 2 (2 3 4 5 )?1 (k1 ) ? B; > : Qr 1 v (i) otherwise 51
cl-c1
where B = (3 4 5 )?1 (k2 ) [ 5?1 (k3 ). This example shows that for any v, Qv and Qr v are equal everywhere but on the union of nitely many sets of the form (1 : : : n )?1 (k) where s1 ; : : : ; sn ; ck are letters of the root term. In that case (1 : : : n )?1(k) is nite since i[is bounded (i n) and 1 : : : n 2 M . We can conclude that the set ?1 (k) is nite and c1 can be set to be its maximum. ck
2 M ;
is a letter of the root term
Claim 3.69 Let be a witness tree with root r, Q be a valuation of the vertex
cl-c2
set. Then (9c2 2 !)(8 vertex v; v0 )(8i 2 !) v v0 ^ (Qr v )(i) 6= (Qr v0 )(i) ) i c2 :
= max Proof Put c2 def
[
h; 0 i2M =
fi 2 ! j (i) 6= 0 (i)g. This de nition is
transparent since the set, we have to take the maximum of is nite. (M = is nite, fi 2 ! j (i) 6= 0 (i)g is nite if 0 .)
Lemma 3.70 Let be a witness tree with root r, Q be a valuation of the vertex
l-c
set. Then (9c 2 !)(8 vertex v; v0 )(8i 2 !) v v0 ^ Qv (i) 6= Qv0 (i) ) i < c: Proof Put c = maxfc1; c2g +1. Claims 3.68 and 3.69 will prove the statement.
Lemma 3.71 Suppose that Ur is a set with cardinality bigger than 2. Then there is a sequence S 2 ! Ur with the property that for all ; 2 [H ] 6 ) Sr 6 Sr To prove this lemma we will need the following claim. Claim 3.72 There is a set D P (!) for that 2 D ) j j = 2, ; 2 D ^ 6= ) \ = ;, ; 2 [H ]; 6 ) jD \ ff(i); (i)g j i 2 !gj = !. Proof Let hi ; ii (i 2 !) be a series of pairs of transformations of ! with the property that for any ; 2 [H ] with 6 we have that hi ; i i = h; i for in nitely many i 2 !. A series of that kind exists since jH j !; jH j !; j[H ]j jH j !; j[H ] [H ]j !. For all k 2 ! we de ne the set k recursively: 52
l-S
cl-h1
0 = f0(i); 0 (i)g for an i 2 ! with 0 (i) 6= 0 (i). [ k = fk (i); k (i)g for an i 2 ! with k (i) 6= k (i) and i 62 ?k 1 ( j ) [ j
In this work, we attempt to alleviate three (more or less) equivalent negative results. These are (i) non-axiomatizability (by any nite schema) of the valid formula schemas of rst order logic, (ii) non-axiomatizability (by nite schema) of any propositional logic equivalent with classical rst order logic (i.e., modal logic of quanti cation and substitution), and (iii) non-axiomatizability (by nite schema) of the class of representable cylindric algebras (i.e., of the algebraic counterpart of rst order logic). Here we present two nite schema axiomatizable classes of algebras that contain, as a reduct, the class of representable quasi-polyadic algebras and the class of representable cylindric algebras, respectively. We establish positive results in the direction of nitary algebraization of rst order logic without equality as well as that with equality. Finally, we will indicate how these constructions can be applied to turn negative results (i), (ii) above to positive ones.
Contents
1 Introduction 2 Results 2.1 2.2 2.3 2.4 2.5 2.6
Preliminaries : : : : : : : : : : : : : : : : : : : : : : : Finitizing algebras of rst order logic without equality Finitizing algebras of rst order logic with equality : : Open Qusetions : : : : : : : : : : : : : : : : : : : : : : Application to logic : : : : : : : : : : : : : : : : : : : : Comparisons with the polyadic notion of a schema : :
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6 8 17 21 21 21
This is a pre-publication version of the paper, nal version will be prepared later and published elsewhere. y [email protected] z [email protected] or [email protected]. Research supported by Hungarian Research Fund, grants F17452, T16448.
1
3 Proofs 3.1 3.2 3.3 3.4 3.5 3.6 3.7
An axiomatization of IGwdfG : : : : : : : : : : : : Theorem 2.7 (1): nite schema axiomatizing IGwdfG Theorem 2.11: compressing the unit element : : : : Quasi-axiomatizing ICdfH : : : : : : : : : : : : : : : Theorem 2.30: the cylindric reduct of Mod(H 1; H 2) Further observations on Mod(H 1; H 2) : : : : : : : : The witness tree method : : : : : : : : : : : : : : : :
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1 Introduction Since it was introduced, rst order logic became the most frequently used and studied logical system. Its logical connectives and basic concepts are motivated by intuitive \mathematical" reasoning. It also plays a fundamental role in logical (formal) investigations of not necessarily mathematical reasoning, cf. e.g. [GG85], [vB91]. First order logic is only one of the logical systems investigated in modern logic. For many logical systems, or logics for short, the methodology of algebraic logic proved to be very useful. This methodology is summarized e.g. in [ANSK] and in [AN94]. Cf. also Blok{Pigozzi [BP89]. The key step in this methodology is that, to any logic L, we associate a class Alg(L) of algebras, called the algebraic counterpart of the logic L. In many cases, investigating Alg(L) is easier than investigating L, e.g. because, in the study of Alg(L), we can use the well developed tools of algebra. A number of problems concerning certain logics L were often solved by translating the problem to Alg(L), solving the algebraic problem by the methodology of algebra, and then translating the algebraic result back to the logical context in which L was originally investigated. Another equally important value of algebraization is obtaining insights into the essential nature of a logic L, via abstraction provided by algebraization. The algebraic counterpart of classical propositional logic is the class of Boolean algebras. Boolean algebras proved immensely useful in studying classical propositional logic, and in understanding its connections with other logical systems. Similarly, for a great number of other logics, studying their algebraic counterparts has proven very useful. One example is propositional modal logic, the algebraic counterpart of which is Boolean algebras with operators. Similar positive examples are most of the non-classical propositional logics, e.g. intuitionistic propositional calculus whose algebraic counterpart is the variety of Heyting Algebras. Cf. e.g. [W88]. In the case of rst order logic, when trying to apply the methodology of algebraic logic in the above outlined fashion, we have to face the following problem. As we said, the algebraic counterpart of classical propositional logic is the class of Boolean algebras. This class turns out to be a nitely axiomatizable equational class. The same is true for the other successful examples we referred to. In each 2
case, Alg(L) is axiomatizable by a nite set of equations or quasi-equations1. As a contrast, in the case of rst order logic, all the algebraic counterparts found before 1987 are very far from being nitely axiomatizable. Namely, J. D. Monk proved in [M70] that the most thoroughly investigated algebraic counterpart of rst order logic with equality, the class RCA! of representable cylindric algebras is not axiomatizable by a nite schema of equations2. A similar result was obtained in Sain{Thompson [SaT91] about the class RQPA! of representable quasi-polyadic algebras (the algebraic counterpart of rst order logic without equality)3 . Monk's theorem was improved by Andreka in [A91]. She proved that any axiomatization of RCA! , besides being in nite, has to be extremely complex. These results motivate the problem of nding an algebraization of rst order logic where the class of algebras associated to this logic is axiomatizable by a preferably nice (simple, transparent, mathematically elegant) schema of equations or quasi-equations. There is a harder version of this problem asking for a strictly nite axiomatization. However, it is understood that a nite schema solution could already be very useful if the schema of axioms is simple enough. This problem is known as the nitization problem in algebraic logic. Actually, as it turns out, it is not a single problem, but rather a family of problems or, perhaps, even a research direction of algebraic logic, cf. e.g. Nemeti [N93], Simon [Si93], Sain [Sa92]. Here we will simply refer to it as the nitization problem. The problem as we formulated above is not concrete enough to be regarded as a tangible mathematical problem. To illustrate this, we note that Simon [Si93] is devoted to showing that many seemingly relevant results are, though mathematically interesting, not even partial solutions of the nitization problem. Therefore, below we recall a suciently concrete formulation of the nitization problem from the literature. The problem of nding a \ nitizable" algebraization of rst order logic was raised in Monk [M70, p.20] and in Henkin{Monk [HM74].4 The formulation in the latter goes as follows: \Devise an algebraic version of predicate logic in which the class of representable algebras forms a nitely based equational class." An equational class is called nitely based if it is axiomatizable by nitely many equations. In [HM74] an algebra is called representable if it is isomorphic to a 1 Quasi-equations are equational implications. 2 Monk obtained similar negative results for RCA3 already in 1964. 3 The Sain{Thompson result was preceded by a relevant result of James Johnson [Jo69]
who proved that RQPAn (2 < n < !) is not nitely axiomatizable. 4 This problem has a long history and many people worked on it, e.g. A. Tarski did so already in the 1940's; but the explicit formulation we would like to use is in the quoted papers of L. Henkin and J. D. Monk. For completeness, we also refer to Henkin{Tarski (1961) [HT61] to the open problem formulated below Theorem 1.14 therein (lines 21{29 on page 96). There they ask for a nite schema axiomatization of Eq(RCA! ). This turns out possible (but only) if we add to RCA! operations like ssuc ; spred de ned herein.
3
subdirect product of set algebras . Therefore the key step is nding an appropriate notion of set algebra. In devising such de nitions, usually, an ordinal is xed in advance, and then the elements of a set algebra are (certain) -ary relations over some set called the base of the algebra. One piece of the connection with logic is that the base of the algebra corresponds to the universe of that model of rst order logic from which it is obtained (see e.g. [HMTII, sections 4.3, 5.6] for details on this connection). Following Jonsson and Tarski, we require that the operations of a set algebra should be invariant under any permutation of the base set5 . An operation with this property is called logical. (This property is called \permutation invariant" in Jonsson [Jo86].) In this work we concentrate on that version of the above quoted problem of Henkin and Monk in which all operations of the set algebras are required to be logical.6 This condition admits the following equivalent formulation. The class of representable algebras can be obtained in the form Alg(L) for some logic L such that, in the model theory of L, isomorphic models satisfy the same formulas | the latter condition is one of the basic axioms of abstract model theory. A solution to the truly nite axiomatization version of the nitization problem was given in Sain [Sa87a] part of which is revised as [Sa87] and abstracted in [Sa92] and [Sa94]. These papers give a complete solution to the case of rst order logic without equality, while the solution there to logic with equality is slightly weaker. Much of the e ort in the just quoted papers goes into ensuring that the number of axioms is strictly nite (as opposed to a nite schema). Therefore the idea comes up that if we require only a nite schema axiomatization (as opposed to \truly nite") then, perhaps, we could achieve substantial improvements in other directions such as simplifying the axioms. Indeed, after [Sa87a] was presented at the Algebraic Logic conference in Asilomar, 1987, both Leon Henkin and J. Donald Monk suggested looking for a solution which, instead of a truly nite set of axioms, provides only a nite schema of axioms, but in such a way that this schema would be simple and easy to work with. This is one of the things we intend to do in the present paper. 7 The question of comparing our notion of a schema with the notion used in Keisler's completeness theorem and in the Daigneault{Monk representation theorem for polyadic algebras (without equality) comes up at this point (see [HMTII] 5.4). We will make such a comparison using results of [NS96] and [Sa95] in Subsection 2.6 at the end of this section. 5 Tarski mentions this requirement already in his 1941 paper on the calculus of relations. See also next footnote. 6 This condition goes back to Lindenbaum{Tarski [LT36], and was extensively discussed in Tarski [Ta87] where it was suggested that the condition is essential for investigations of truly logical nature. Further considerations for necessity of requiring all the operations to be logical (i.e. permutation invariant) are in [Sh91] and [vB96] 3.3.2 (pp.67{68). It was proved e.g. in Biro [Bi89] that permutation invariance is a necessary condition in the nitization problem. Cf. also Nemeti [N91]. 7 A related open problem is to simplify the axiomatization given in the original papers (e.g. in [Sa87]).
4
A further aim of this paper is to improve the positive results to a tighter or more orthodox class of set algebras. This tighter class is denoted below as CdfG . To rst order logic without equality, we will associate a fairly simple and powerful class CdfG of set algebras. A CdfG algebra A is a subalgebra of the Boolean algebra of subsets of the Cartesian space ! U expanded with some extra operations corresponding to quanti cation and substitution8 . We will axiomatize CdfG by a relatively simple nite schema of quasi-equations (cf. Theorem 2.5). A modi ed version of this class will receive an equational axiomatization. It remains an open problem to simplify these axiomatizations. To the same logic we will also associate a slightly bigger class GwdfG of set algebras. We will be able to prove more results about this second class (and also we will be able to simplify the axioms for the latter). CdfG will meet all the requirements of the (schema version of the) nitization problem quoted way above. E.g., its operations will be logical and natural set theoretic operations on relations. The same applies to GwdfG . To rst order logic with equality, we will associate a class Cssuc;pred of set algebras, which has the following positive properties. The equational class V generated by Cssuc;pred is axiomatizable by a nite schema of equations, which is even simpler than the one we give to logic without equality. Further, Cssuc;pred is an expansion of cylindric set algebras with some extra operations. These extra operations are again logical and simple set theoretic ones, just as in the previous case. Now, our nite schema axiomatizable V has the property that if we omit the extra operations from its members then we get exactly the class RCA! of representable cylindric algebras. Let us see what this means in more detail. To the natural algebraic counterpart RCA! of rst order logic with equality, we add simple logical operations in such a way that, in the language of the so expanded class, we can write up a nite schema of axioms which completely describes the original operations of RCA! , but not necessarily the new operations themselves. More concretely, any model A of our nite schema is representable as a true set algebra in the following sense. A is isomorphic to some B, where the elements of B are real relations, and all the old cylindric algebraic operations are the usual set theoretic ones on B, but perhaps the new operations of B remain \abstract". At this point it might be of interest to note that, in the book [TaG87] (on page 62), the nitization problem was formulated in this weaker sense which we just described in connection with logic with equality. That is, Tarski and Givant ask for expanding the existing algebras with new logical operations in a way that there would be a nite set of axioms in the expanded language which 8 These algebras are often called \square algebras".
5
describes the old operations completely, but not necessarily the new ones. For a comparison of the two versions of the nitization problem see the series of remarks in [N93] beginning with Remark 2. Summing up, we can give a solution to the more \ambitious" formulation of the schema version of the nitization problem for logic without equality, while for logic with equality we could solve only the above outlined Tarski-Givant style formulation of the nitization problem. When de ning the nite schema axiomatizable ( nitizable for short) algebraic counterparts GwdfG; Cssuc;pred of rst order logic, one has a certain freedom of choice concerning exactly how these algebras will look like. We will explore this freedom of choice in a metatheory, asking ourselves the question which of these choices of GwdfG ; Cssuc;pred are workable, and which are already non- nitizable. We would like to note that our approach to the case with equality is strongly related to William Craig's pioneering work summarized, e.g., in his book [Cr74]. Actually, our work can be viewed as a continuation of the approach initiated by Craig. *** The formulation of the results mentioned above together with the basic de nitions can be found in Section 2. In detail, Subsection 2.1 contains preliminary material for the rest of the paper, while Subsections 2.2 and 2.3 are devoted to the nitization of rst order logic without equality and with equality, respectively. Subsection 2.5 concerns applications to `pure' logic. The proofs are given in Section 3. At the end of the paper there are further historical remarks. Any proof which would rely on references not available to the reader (or would seem incomplete for any other reason) is available with all such missing data added from the authors.
2 Results
2.1 Preliminaries
Throughout this paper, ! denotes the set of all natural numbers. If A and B are arbitrary sets then A B denotes the set of all functions from A into B . Thus, for any set U , ! U denotes the set of all !{sequences over U . We often call the elements of ! ! transformations of !. A function f 2 ! ! is called a nite transformation of ! if f (i) = i for all but nitely many i 2 !. Examples of nite transformations of ! are transpositions and replacements, which are de ned as follows. For each i; j 2 !, the transposition [i; j ] interchanges i and j and leaves all the other elements of ! xed; the replacement [i=j ] maps i to 6
j and leaves all the other elements of ! xed. It is easy to see that the set of all nite transformations of ! forms a semigroup with function composition as the semigroup operation. Clearly, this semigroup is generated by the set of all replacements and transpositions. Examples of transformations of ! that are not nite are the successor suc and the predecessor pred de ned as follows. For every i 2 !, suc(i) = i + 1 and i ? 1 if i 6= 0 . pred(i) = i if i = 0 If H is a set then H denotes the set of sequences of elements of H . If T is a subset of ! ! then [T ] denotes the subsemigroup of h! !; i generated by T . For any class K of similar algebras, IK , HK , SK , PK and UpK denote, respectively, the classes of isomorphic copies, homomorphic images, subalgebras, isomorphic copies of direct products and isomorphic copies of ultraproducts of members of K . SirK denotes the class of subdirectly irreducible members of K. Let t and t0 be similarity types with t0 t. For any class K of algebras of similarity type t, Rdt0 K denotes the class of algebras of similarity type t0 de ned as follows. A 2 Rdt0 K if and only if A is the t0 {reduct of some algebra in K . Informally speaking, we obtain the t0 {reduct of an algebra by deleting those operations of it that are not in t0 . For any set of rst order formulas of similarity type t, Modt () denotes the collection of those models (of type t) in which is valid. When t is known from context, we will often write Mod() instead of Modt (). For a class of algebras K , Eq(K ) denotes the set of equations valid in K while Qeq(K ) is the set of valid quasi-equations of K . If f is a function and U is a subset of its domain then f dU denotes the restriction of f to U . For the pair of sets U; V , U ! V expresses the relation that U is a nite subset of V . As we outlined in the Introduction, we will introduce several (new) kinds of set algebras. These algebras will be de ned to be Boolean algebras of relations augmented with some extra operations. The greatest element of a Boolean algebra (with perhaps some extra operations) is called its unit. Therefore, when speaking about a class K of special Boolean algebras, then it is meaningful to talk about the class of unit elements of members of K . These will be called K {units. Of these classes of set algebras, our rst theorem (Theorem 2.5) will involve only the most important one called CdfG . Actually, one of the main results of this paper is Theorem 2.5 (i), (ii). To understand that result it is enough to read only a small fraction of the de nitions below. Namely, after reading De nition 2.1 (i), the CdfH part of De nition 2.2 and De nition 2.4 the reader can go directly to read Theorem 2.5. 7
df-unit
De nition 2.1 ([HMTII] Def.3.1.2)
Let U be an arbitrary set. (i). The set ! U is called the Cartesian space with base U (and dimension !). (ii). By a generalized Cartesian space we mean a disjoint union of Cartesian spaces. That is, a set [ is called a generalized Cartesian space (of dimension !) if it has the form ! Ui , where I is a set, and for all distinct i; j 2 I , i2I
Ui \ Uj = ;. (iii). For every p 2 ! U , we set ! U (p) = fq 2 ! U :
fi 2 ! : q(i) 6= p(i)g is niteg;
and we call ! U (p) the weak Cartesian space (of dimension !) determined by p. U is called the base of ! U (p) . (iv). By a generalized weak Cartesian space or Gws! {unit we mean a disjoint union of weak Cartesian spaces. That is, a set is called [ a generalized weak Cartesian space of dimension ! if it has the form ! Ui(pi ) for some sets
i2I I; Ui and functions pi 2 ! Ui (i 2 I ) such that for all distinct i; j 2 I , ! U (pi ) \ ! U (pj ) = ;. i j [ (v). We say that a Gws! {unit ! Ui(pi ) is normal if for all distinct i; j 2 I , i2I Ui and Uj are either disjoint or equal. [ (vi). Finally, we say that a Gws! {unit ! Ui(pi ) is compressed if Ui = Uj for i2I all i; j 2 I .
2.2 Finitizing algebras of rst order logic without equality
De nition 2.2 Suppose that H is a xed subset of ! !. An algebra is called a full generalized weak H {cylindric set algebra with unit V if it has the form
hP (V ); [; ?; s ; ck i 2H;k2! ; where
V is a Gws! {unit, for every 2 H and x V , s (x) = fq 2 V : q 2 xg, 8
d-igwsg
s (V ) = V whenever 2 H , ck is the k-th cylindri cation, that is, for any x V , ck (x) = fq 2 V : (9q0 2 x)(8i 2 !) i =6 k ) q(i) = q0(i)g: If A is a full generalized weak H {cylindric set algebra with unit V , then A will often be denoted by P (V ), if H is known from context. By a generalized weak H {cylindric set algebra (or brie y, by a GwdfH ) we mean a subalgebra of a full generalized weak H {cylindric set algebra. The class of all generalized weak H {cylindric set algebras is denoted by GwdfH 9 . The similarity type of GwdfH is denoted by tH . The classes of all algebras of GwdfH , whose unit elements are a Cartesian space, a generalized Cartesian space, a normal Gws! {unit or a compressed Gws! {unit, are denoted by CdfH , GdfH , GwdfHnm or GwdfHcm respectively.
De nition 2.3 Let H ! !, V be a generalized Cartesian space10, x V . We de ne the operator c(!) : P (V ) ! P (V ) as
d-111
c(!)x def = fs 2 V : (9q 2 c0 x) s(0) = q(0)g: A = hBec; c(!) i is called GdfHec i B is a GdfH with unit V . An A 2 GdfHec is
called CdfH if its unit element is a Cartesian space.
To help the intuition we point out that IGdfHec = SPCdfHec while in CdfHec c(!)x = 1 i x > 0 and c(!)0 = 0. Adding the new operator c(!) to GdfH amounts to adding the formation of existential closure ' ! 9x' to the logic being algebraized (where x contains all the variables occuring in '). The abbreviation `ec' in GdfHec (and in CdfHec) refers to this fact.
De nition 2.4 Throughout this paper, G denotes the following set.
d-g
G def = f[i=j ] : i; j 2 !g [ fsuc; predg: The following theorem summarizes the main results of this section. 9 Gwdf refers to the name `generalized weak diagonal free cylindric set algebra' used in
[HMTI] to denote the H -free reduct of our class. 10 The de nition makes perfect sense for the case when V is only a generalized weak Cartesian space (or even any subset of ! U for some set U ). For brevity, we will not discuss here the more general cases.
9
csicsa1
Theorem 2.5 (i). ICdfG is a nite schema axiomatizable quasi-variety. That is, there is a nite schema QxG 11 of quasi-equations such that ICdfG = Mod(QxG) :
(ii). IGdfGec is a nite schema axiomatizable discriminator variety. (iii). HCdfG is a variety axiomatized by a nite set AxG 12 of equation schemas. Thus, AxG axiomatizes Eq(CdfG ) i.e. CdfG j= e i AxG ` e; for any equation e in the language of CdfG . The statements of this theorem are restated in Theorems 2.7, 2.15, 2.16 below in a more detailed form. In the following part of this section we present the nite sets of equations and quasi-equations that axiomatize HCdfG , ICdfG and IGdfGec. Further, the related weak classes are examined and a detailed characterization is given. (Theorem 2.7) Finally, we show that many of the results remain valid if we replace the set G by an arbitrary set H of transformations provided that H satis es certain properties. (Theorems 2.11, 2.19, 2.20) De nition 2.6 We de ne the nite set AxG of axiom schemas to be the union of sets (1){(6) below. (1). A nite set of equations axiomatizing Boolean algebras. (2). For every 2 G, the following axioms stating that s is a Boolean homomorphism: s (x _ y) = s (x) _ s (y); s (?x) = ?s (x): (3). The axioms governing s[i=j] 's: for all distinct i; j; k 2 !, s[j=i] s[i=j] x = s[j=i] x; s[j=k] s[i=j] x = s[j=k] s[i=k] x: (4). The axioms describing the connections between s[i=j] 's, ssuc and spred: spredssucx = x; ssucspredx = s[0=1]x; ssucs[i=j] x = s[suc(i)=suc(j)] ssucx for all distinct i; j 2 !. (5). Two axiom schemas about ci and the Boolean operations: for all i 2 !, x ci x, ci (x _ y) = ci x _ ci y: (6). The axiom schemas describing the connection between s[i=j] 's and ck 's: for all distinct i; j; k 2 !, s[i=j] ci x = cix; cis[i=j] x = s[i=j] x; ck s[i=j] x = s[i=j] ck x: 11 QxG is presented in De nition 2.14 below. 12 AxG is presented in De nition 2.6 below.
10
d-ax
Theorem 2.7 (characterization of IGwdfG and related classes)
th-main
Let G and AxG be as in De nitions 2.4 and 2.6. Then statements (1){(4) below hold. (1). IGwdfG is axiomatizable by the nite set AxG of equation schemas i.e. IGwdfG = Mod(AxG). (2). IGwdfG is a variety. (3). IGwdfG = IGwdfGnm = IGwdfGcm = HCdfG . (4). The set Eq(CdfG) of equations is axiomatizable by the nite set AxG of equation schemas i.e.
CdfG j= e i AxG ` e; for any equation e in the language of CdfG .
Proof (2) is a corollary of (1) while (1) is proved in Subsections 3.1 { 3.3.
To prove (3) we note that statement (3) is a special case of Theorem 2.11 (ii), (iii). To see that the later is applicable, we need to show that the successor, the predecessor and all the substitutions are bounded transformations of ! and they have right quasi-inverses in [G]. (Consult De nition 2.9 for the notion of bounded, quasi-bounded and quasi-invertible transformation.) It easy to verify that the given transformations have the required properties using facts as suc pred(i) = pred suc(i) = i whenever i > 0. Finally, (4) follows from (3) and (1).
Later, in Remark 2.22, we show that for a slightly modi ed version of IGwdfG (where only the 0-th cylindri cation is a basic operation, the other cylindri cations are terms of the reduced language) similar results can be achieved. In the proof of the previous theorem we referred to a metatheorem (2.11). There we show that if H satis es certain requirements then the classes IGwdfH and IGwdfHcm actually coincide with IGwdfHnm.
Notation 2.8 We will often deal with pairs of sequences that agree on all but
nt-approx
nitely many coordinates. If q1 and q2 are sequences then
q1 q2 i jfi 2 ! : q1 (i) 6= q2 (i)gj < !: Clearly, is an equivalence relation.
De nition 2.9 Let f; g 2 ! ! be arbitrary. 11
d-bounded
f is said to be bounded if it does not send in nitely many elements of ! into the same place, that is, if jf ?1(i)j < ! for all i 2 !. f is said to be quasi-bounded if there is a k 2 ! for which jf ?1 (f (k))j < !. g is called the right (left) quasi-inverse of f if f g Id (g f Id), where Id is the identity transformation of !. f is said to be right (left) quasi-invertible in H ! ! if it has a left (right) quasi-inverse in H .
De nition 2.10 (rich semigroup) Suppose that i; j; k 2 !, A !, f; g 2 ! !. Then f [i=j ] and f [A=g] are transformation of ! de ned as
f [i=j ](k) =
j if k = i f (k) otherwise
f [A=g](k) =
d-rich
g(k) if k 2 A f (k) otherwise.
(Note that f [i=j ] and f [i=j ] are di erent transformations.)13 A semigroup Q ! ! is called rich if it satis es the following two conditions: (8i; j 2 !)(8f 2 Q) f [i=j ] 2 Q, (9s^; p^ 2 Q)(8f 2 Q) p^ s^ = Id ^ Rng(^s) 6= ! ^ (^s f p^)[(! ? Rng(^s))=Id] 2 Q.
Theorem 2.11 Let H be an arbitrary set of transformations. (i). IGwdfHnm = HSPCdfH is a variety. Thus Eq(CdfH ) = Eq(GwdfHnm). (ii). If for all 2 H , is bounded then IGwdfHnm = IGwdfHcm = HCdfH : (iii). If for all 2 H , is quasi-bounded and right quasi-invertible in [H ], then
IGwdfH = IGwdfHnm: (iv). Assume that [H ] is a nite schema presented14 rich semigroup. Then, IGwdf nm is axiomatizable by a nite schema of equations.
(i) and (iv) are quoted from [Sa92] (Theorem 2.1 (2)), the proof of (ii) and (iii) can be found in Subsection 3.3. The example below shows that the requirement given in Theorem 2.11 (iii) cannot be omitted. In some cases Eq(GwdfH ) 6= Eq(GwdfHnm ). 12
th-nc
Example 2.12 Let doub 2 ! ! be de ned by doub(i) = 2i for every i 2 !, let H = fdoubg and let e be the equation c0 sdoub x = sdoubc0 x. Then GwdfHnm j= e but GwdfH 6j= e: Proof Let V = ! 2(h0;1;0;1;:::i) [ ! 3(h0;0;0;0;:::i). Then P (V ) 2 GwdfH n IGwdfHnm since e fails in P (V ) (choose x = fh2; 0; 0; 0; : : :ig) but e is satis ed in every algebra of IGwdfHnm. We turn our attention to the study of classes of algebras with square (Cartesian space) unit elements. We prove that the class ICdfG is a nite schema axiomatizable quasi-variety and is strictly smaller than IGwdfG .
ex0
Example 2.13 Let q be the quasi-equation spredx = ?x ) 0 = 1. Then15 GdfG j= q but IGwdfG 6j= q: Proof In a G{cylindric generalized set algebra we can pick a constant sequence s in the universe. For any element x of the algebra, the chosen s is in x if and only if it is in spredx. It implies that q is in the symmetric di erence of ?x and spredx so spredx =6 ?x. On the other hand, if we put V = ! 2(h1;0;1;0;:::i) [ ! 2(h0;1;0;1;:::i) then e fails in P (V ). (Put x to be ! 2(h0;1;0;1;:::i) and for that x, spredx = ?x.)
ex1
De nition 2.14 (i). Let n 2 !. We de ne c(n)x def = c0 c1 : : : cn?1 x. Furdef ther, sn-pred x = spreds[0=1]s[1=2] : : : s[n?1=n] x. Hence, sn-pred is a term
d-qx1
function16 .
13 f [i=j ] can be distinguished from f [A=g] by noticing that always jgj ! > i. 14 We use the notion of a nite schema presented semigroup in the usual sense. For com-
pleteness we will recall this concept in De nition 3.7 in the `Proofs' section. 15 For completeness we note connections of Example 2.13 and the theory of polyadic algebras (PA! 's) and PA's with equality (PEA! 's) without recalling these two classes. We use the notation of [HMTII]. Although AxG 6j= q, we have PA! j= q (e.g. by the Daigneault{Monk theorem). On the other hand, there are equations e such that PEA! 6j= e but RPEA! j= e, cf. Nemeti-Sagi [NS96]. 16 In set algebras, the term function sn-pred coincides with s for = h0; 1; : : : ; n ? 1; n ? 1; n; n + 1; : : :i, that is, n if i < n (i) = (predd! ? n) [ (Iddn) = ipred(i) otherwise.
13
(ii). We de ne QxG to be the following schema of quasi-equation.
[
im
c(ni)xi = 1 ^
for n0 ; n1 ; : : : ; nm ; m 2 !.
Theorem 2.15 Proof
17
^
im
sni-predxi ?xi ) 0 = 1;
cj1
ICdfG = Mod(AxG [ QxG):
The general Theorem 2.19 should be applied to H = G. Qx can be simpli ed to QxG and Theorem 2.7 implies that Eq(IGwdfG ) follows from AxG . It is left to the reader to show the rst implication.
We conjecture that QxG can be considerably simpli ed. Cf. e.g. the de nition of Ax2 and proof of Proposition 1 in Andreka [A91] part II. ec Theorem 2.16 IGdf G is a nite schema axiomatizable discriminator variety. ec Furthermore, IGdfG = Mod() where is AxG together with the following set of axiom schemas: (i). axioms stating that c(!) is a complemented closure operator, i.e. x c(!) x, c(!)(x [ y) = c(!)x [ c(!)y, c(!)c(!)x = c(!)x, c(!)?c(!)x = ?c(!)x, (ii). axiom schemas stating that c(!) majorizes all the other operators, i.e. c(!)x f (x) for all f 2 fci; s : i 2 !; 2 Gg, (iii). axioms corresponding to[QxG : for all n0 ; n1 ; : : : ; nm; m 2 !, [ ?c(!) ?( c(ni ) xi ) c(!) (xi \ sni -pred(xi )).
ICdf ec
im
t-nemeti
im
ec G is the class of subdirectly irreducible members of IGdfG .
Theorem 2.16 is a special case of the general Theorem 2.20. Open question 2.17 Find simpler axiom schemas aximatizing ICdfG and IGdfecG, respectively. Below we show that ICdfH is a quasi-variety, provided that H satis es some properties. Furthermore, we give a set Qx of quasi-equations that axiomatize ICdfH over IGwdfH . Results 2.19 and 2.20 were obtained jointly with Istvan Nemeti. De nition 2.18 (i). Let n 2 !, ? = f 0; : : : ; n?1g !. We recall from [HMTI] that c(?) x def = c 0 : : : c n?1 x.
17 In [Sa87] a simple proof is given to the fact that ICdfG is closed under SPUp, i.e. it forms a quasi-variety. Finite schema axiomatizability is not proved there.
14
ques1
d-qx2
(ii). We de ne Qx to be
[
im
c(?i)xi = 1 ^
^ \ im 2Hi
s (x) = 0 ) 0 = 1;
for m 2 !; ?0 ; : : : ; ?n?1 ! !, Hi ! [H ] such that ( 2 Hi ) d? = i Idd? ). i
Theorem 2.19 Let H ! ! such that [H ] contains all nite transformations
t-csax
and every element of H is bounded, left and right quasi-invertible in H . Then (1). ICdfH is a quasi-variety, (2). ICdfH is axiomatized by Qx over GwdfH , that is, Qx [ Eq(GwdfH ) axiomatizes ICdfH .
(1) is a consequence of (2) while (2) is proved in Subsection 3.4.
Theorem 2.20 Assume that H ! ! is such that ICdfH is a quasi-variety. Then IGdfHec is a discriminator variety whose subdirectly irreducible members are exactly the elements of ICdfHec . If all the assumptions of Theorem 2.19 hold for H then IGdfHec is axiomatized over IGwdfH by the axioms (i) and (ii) of Theorem 2.16 together with (iii'). axioms corresponding to Qx: for all m 2 !; ?0; : : : ; ?n?1 ! !, Hi ! [H ] such that ( 2 Hi ) d? = Idd? ) we state [ [ i \ i ?c(!) ?( c(?i ) xi ) c(!) ( s x). im
im
2Hi
Proof Let be theecset equations validecin GdfHec, that is, = Eq(GdfHec). We want to prove IGdfH = Mod(). IGdfH Mod() is obvious. For proving the other direction, it is enough to show that
Sir(Mod()) ICdfHec IGdfHec: Therefore, assume A 2 Sir(Mod()). A j= x 6= 0 ! c(!) x = 1 can be veri ed as follows. Suppose for contradiction that c(!) x = 6 1 for some element x of A. ec
Since axioms (i) and (ii) certainly hold in GdfH , they are in . But using axioms (i) and (ii) we can prove that the relativization with x and that with ?x decomposes A to the ^ subdirect product of the two images. Let A = hB; c(!) i. If i = i ! = is a quasi-equation valid in CdfH iFrom the main result of Thompson [To93] we know that [1=0]n 0 [1=0]m = [1=0]n 00 [1=0]m imply ` [0=1]n 0 [0=1]m = [0=1]n 00 [0=1]m. Collecting all the information we have: ` sucn predm = sucn predm . But this implies ` predn sucn predm sucm = predn sucn predm sucm . Using (3.1) again we can conclude ` = and this completes the proof. The set of de ning equations listed in 3.8 is not independent. Below we show that equations (3.4), (3.5), (B6), (B7) are implied by the others.
Claim 3.9 (3.1),(3.2),(3.3),(B1),(B3),(B4),(B5)`(B6) Proof First we prove that the listed B-axioms imply [i=j ][j=k][k=i][i=j ][a=b] = [i=k][k=j ][j=i][a=b] where all i; j; k; a; b are distinct: [i=j ][j=k][k=i][i=j ][a=b]
B=1
B=4
B=1
B 3;B 4
= = B=3
B 5;B 3;B 4;B 1
26
[i=j ][j=k][k=i][i=j ][a=j ][a=b] [i=j ][j=k][k=i][a=j ][i=j ][a=b] [i=j ][j=k][k=i][a=i][a=j ][i=j ][a=b] [i=j ][j=k][a=i][k=a][a=j ][i=j ][a=b] [a=j ][i=j ][j=k][i=j ][k=a][a=b] [a=j ][i=j ][j=k][i=k][k=a][a=b]
cl-b7
B 4;B 1;B 3
=
B=5
B 3;B 4;B 5
= B 1;B 5 =
[a=j ][i=k][j=i][k=a][a=b] [i=k][a=j ][k=a][j=i][a=b] [i=k][k=j ][a=j ][a=b][j=i] [i=k][k=j ][j=i][a=b]:
Now we can compute: [i=j ][j=k][k=i][i=j ] 3=:1 pred suc[i=j ][j=k][k=i][i=j ] 3:3;3=:1;3:2 pred[i + 1=j + 1][j + 1=k + 1][k + 1=i + 1][i + 1=j + 1][0=1]suc = pred[i + 1=k + 1][k + 1=j + 1][j + 1=i + 1][0=1]suc 3:3;3=:1;3:2 [i=k][k=j ][j=i].
Claim 3.10 (3.1),(3.2),(3.3),(B1),(B3),(B4),(B5)`(B7) Proof First we prove that the listed B-axioms imply
cl-b8
[i=k][k=l][l=j ][j=i][i=l][a=b] = [i=j ][j=k][k=l][l=i][a=b] where all i; j; k; l; a; b are distinct: [i=k][k=l][l=j ][j=i][i=l][a=b]
B 1;B 4
= = B 4;B 5;B 3 = B 4;B 3 = B 5;B 1 = B=5 B 3;B 4;B 5;B 1 = B 1;B 5 = B 1;B 4;B 3
[i=k][k=l][l=j ][j=i][a=l][i=l][a=b] [i=k][k=l][l=j ][a=i][j=a][a=l][i=l][a=b] [a=k][i=k][k=l][l=j ][i=l][j=a][a=b] [a=k][i=k][k=l][i=j ][l=i][j=a][a=b] [a=k][i=j ][k=l][l=i][j=a][a=b] [i=j ][a=k][j=a][k=l][l=i][a=b] [i=j ][j=k][a=k][a=j ][k=l][l=i][a=b] [i=j ][j=k][k=l][l=i][a=b]:
Now we can compute: [i=k][k=l][l=j ][j=i][i=l] 3=:1 pred suc[i=k][k=l][l=j ][j=i][i=l] 3:3;3=:1;3:2 pred[i + 1=k + 1][k + 1=l + 1][l + 1=j + 1][j + 1=i + 1][i + 1=l + 1][0=1]suc = pred[i + 1=j + 1][j + 1=k + 1][k + 1=l + 1][l + 1=i + 1][0=1]suc 3:3;3=:1;3:2 [i=j ][j=k][k=l][l=i].
Claim 3.11 (3.1),(3.2),(3.3),(B3),(B4)`(3.5) 27
cl-35
Proof [0=j ]pred 3=:1 pred suc[0=j ]pred 3=:3 pred[1=j + 1]suc pred 3=:2
pred[1=j + 1][0=1] B3=;B4 pred[0=j + 1][1=j + 1]
Claim 3.12 (3.1),(3.2),(3.3),(B4),(B5)`(3.4)
cl-34
Proof [i=j ]pred 3=:1 pred suc[i=j ]pred 3=:3 pred[i + 1=j + 1]suc pred 3=:2
pred[i + 1=j + 1][0=1] B4orB = 5 pred[0=1][i + 1=j + 1] 3=:2 pred suc pred[i + 1=j + 1] 3=:1 pred[i + 1=j + 1] We quote an important result of Pinter.
Claim 3.13 (Lemma 2.2 of [P73]) For every distinct i; j; k; l 2 ! we have AxG ` s[i=k] s[i=j] = s[i=j] ; AxG ` s[k=j] s[i=j] = s[i=j] s[k=j] ; AxG ` s[k=l] s[i=j] = s[i=j] s[k=l] : We will need the following technical observation that originates from Sandor Csizmazia:
Lemma 3.14 AxG together with the equations (*3), (*4), (*5), (*1) without ci cj x = cj cix implies that for any 2 f[i=j ] j i; j 1g [ f[1=0]suc; pred[1=2]g s c0x = c0s x Proof There is nothing to prove in s[i=j] c0x = c0s[i=j] x (i; j 1) since it is
included in AxG . It was formulated in the lemma only for further applications. To prove the other two statements we will need the following claim quoted from Pinter [P73]:
z
B
}|
{
cl-p
lm-uj7
cl-ci
Claim 3.15 cix = min fy j y = s[i=j] y; y xg Proof ci x 2 B since s[i=j] ci x = ci x and cix x that is (5) and (6) of AxG. 1 For any y 2 B y ci x since ci x ci y = ci s[i=j] y =7 s[i=j] y = y. Further observations are needed.
cl-defci
Claim 3.16 cix = s[0;i]c0 s[0;i]x Proof Using the previous claim, we have to prove that s[0;i]c0 s[0;i]x 2 B def = fy j y = s[i=j] y; y xg and y 2 B ) y s[0;i] c0 s[0;i] x. s[i=j] s[0;i]c0s[0;i]x =3 s[0;i]s[0=j] c0s[0;i]x =6 s[0;i]c0s[0;i]x: 28
(5) implies c0 s[0;i] x s[0;i] x, using (2) we have s[0;i] c0 s[0;i] x s[0;i] s[0;i] x =4 x. 1;2 y 2 B ) y x ) s[0;i] c0s[0;i]x s[0;i]c0s[0;i]y y=2B s[0;i]c0s[0;i]s[i=j] y =3 s[0;i]c0s[0=j] s[0;i]y =6 s[0;i] s[0=j] s[0;i]y =3 s[0=j] y y=2B y: cl-c1s01
Claim 3.17 c0s[1=0]x = c1s[0=1]x Proof c0s[1=0]x =3 c0s[1=2]s[1=0]x =6 s[1=2]c0 s[1=0]x =3 s[1=0]s[1=2]c0s[1=0]x =6 s[1=0]c0s[1=2]s[1=0]x =3 s[1=0]c0s[1=0]x =3 s[0;1]s[0=1]c0s[0;1]s[0=1]x =6 s[0;1]c0s[0;1]s[0=1]x Cl:=3:16 c1s[0=1]x Claim 3.18 ci+1ssucx = ssucci x Proof Let B1 = fy j s[i+1=i+2] y = y; y ssucxg; B12 = fy j s[i+1=i+2] y = s[0=1] y = y; y ssucxg; B21 = fssuc y j s[i=i+1] y = y; y xg; B2 = fy j s[i=i+1] y = y; y xg: The following statements prove this claim: I ci+1 ssuc x = minB1 ;
. II minB1 = minB12 ; III B12 = B21 ; IV minB21 = ssuc minB2 ; V minB2 = ssuc ci x: I, V follow from Claim 3.15, IV from the observation that y 2 B2 $ ssucy 2 B21 . Similarly to the proof of the previous claim, in II we need to show that B12 B1 (trivial) and (8y 2 B1 ) ? c0 (?y) 2 B12 ^ ?c0 (?y) y. So suppose y 2 B1 . Then s[i+1=i+2] (?c0 (?y)) Ax = ?c0 (?s[i+1=i+2] y) = ?c0 (?y), s[0=1](?c0(?y)) = ?s[0=1]c0(?y) = ?c0 (?y) and y ssuc x ) ?c0 (?y) ?c0 (?ssucx) = ?c0 ssuc(?x) = ?ssuc (?x) = ssucx. In * we used that ssuc x = ssucspredssucx = s[0=1] ssucx = c0 s[0=1] ssuc x = c0 ssucx. Turning to III, suppose that y 2 B12 . Then y = s[0=1] y = ssuc (spredy). s[i=i+1](spredy) = spredssucs[i=i+1](spredy) = spreds[i+1=i+2] ssucspredy = spreds[i+1=i+2] s[0=1]y = spredy, y ssuc x ) spredy spredssucx = x. On the other hand if we suppose that ssucy 2 B21 then s[i+1=i+2] ssucy = ssucs[i=i+1] y = ssucy, s[0=1]ssucy = ssucs[ssuc=y] = ssucy and y x ) ssucy ssuc x.
Now we can prove the two statements formulated in the lemma: c0 s[1=0]ssucx = ssucc0 x =6 ssucs[0=1]c0x =4 s[1=2]ssucc0x =3 29
cl-cisuc
=3 s[1=0] s[1=2] ssucc0 x = s[1=0] ssuc c0 x, c0 spreds[1=2]x =4 spredssucc0 spreds[1=2]x = spredc0s[0=1]ssucspreds[1=2]x = = spredc0 s[1=2] x = spreds[1=2] c0 x. In * we used c0 s[1=0] ssucx Cl:=3:17 c1 s[0=1] ssucx Ax = c1 ssucspredssucx Ax = Ax Cl: 3:18 = c1 ssuc x = ssucc0 x.
Proof of Theorem 2.7 (1) Using the result of the previous subsection stated in Corollary 3.6 we need to prove only that AxG `(*1){(*7) and AxG ` ci x = s[0;i]c0s[0;i]x. (*2) is a trivial consequence of (2). (3) and (4) together imply (*3) and (*4) since if A is an algebra of IGwdfG and A is the semigroup of all polynomials of A, expanded with the constant symbols [i=j ]; suc; pred evaluated into the polynomials s[i=j] ; ssuc; spred then A is a model of (3.1){(3.6) ((3) and (4) is the relevant part of the set of equations (3.1){(3.7) | see Claim 3.9{3.13 | written in the language of A ) and so = forces A j= = and that is s x = s x. (*7) is included in (6). To prove (*5), suppose and are in G and and agree everywhere but in i. Then [i=i + 1] and [i=i + 1] are equal everywhere. s ci x =6 s s[i=i+1]ci x = s [i=i+1]ci x = = s [i=i+1] ci x = s s[i=i+1] ci x =6 s ci x In (*1) x ci x, ci (x _ y) = ci x _ ci y follow from (2) and (5). ci ci x = ci x and ci (?ci x) = ?ci x is a consequence of (2) and (6): ci ci x = ci s[i=i+1] ci x = s[i=i+1]ci x = ci x and ci(?cix) = ci (?s[i=i+1] cix) = ci s[i=i+1] (?ci x) = = s[i=i+1] (?ci x) = ?s[i=i+1] ci x = ?ci x To prove (*6) we will need the following observation: If T0 = ff 2 [G] j f ?1 (0) = f0gg and G0 = f[i=j ] j i; j 1g [ f[1=0]suc; pred[1=2]g then T0 is generated by the meanings of the elements of G0 . To verify this statement de ne the function 0 mapping [G] into itself
f 0 = (suc f pred)[0=0] (Since [G] is rich f 0 2 [G].) It is easy to check that 0 is a homomorphism ((f g)0 = f 0 g0 ), furthermore, the 0 image of [G] is T0 (if f 2 [G] then f 0?1 (0) = f0g so f 0 2 T0 and if g 2 T0 then (pred g suc)0 = g). Finally, we note that the set of transformations named by the elements of G0 coincide with the 0 image of the set of those transformations that are named by the elements of G. We note that for any 2 G0 , s c0 x = c0 s x. This follows from Lemma 3.14 since all the extra equations listed in the formulation of the lemma are 30
consequences of AxG , as it was proved up till this point. In passing, we mention that the same reasoning shows that Claim 3.16 proves that the de nition of ci (i > 0) is implied by AxG . Turning our attention again to (*6), suppose 2 G , ?1 (j ) = fig. Then f = [j; 0] [i; 0] 2 T0 . Using the previous observation, f is a composition of functions named by elements of G0 say, f = 1 : : : n . (3), (4) and (6) imply that s[j;0] [i;0]c0x = s1:::n c0x = s1 : : : sn c0x = c0s1 : : : sn x = c0 s[j;0] [i;0]x: Finally we derive
s ci x =y s s[i;0]c0s[i;0]x = s[j;0]s[j;0] [i;0]c0s[i;0]x = = s[j;0] c0 s[j;0] [i;0]s[i;0] x =y cj s x: (In y we used that the de nition of ci and cj is already proven from AxG .) It is left to prove ci cj x = cj ci x in (*1). For that we need x y ) ci x ci y and ci cj ci x = cj ci x. ci cj ci x =6 ci cj s[i=k] ci x =6 ci s[i=k] cj ci x =6 s[i=k] cj ci x =6 cj s[i=k] ci x =6 cj ci x. Now ci cj x ci cj ci x = cj ci x and symmetrically cj ci x ci cj x giving cj ci x = ci cj x. In the previous two subsections we distinguished the symbol from the transformation denoted by it. This distinction is not necessary anymore since axioms (*3), (*4) of De nition 3.3 hold in all algebras studied hereafter. So we return to the original notation where denotes both the symbol and the corresponding transformation hoping that context will help.
3.3 Theorem 2.11: compressing the unit element
Proof of Theorem 2.11cm (ii) IGwdfHcm = HCdfH can be veri ed as follows. Suppose that A 2 GwdfH . Put V to be the unit element, A the universe of A.
Since V is a compressed Gws! {unit, every weak space in V has the same base, say U . Put B = fx ! U : x \ V 2 Ag. Clearly, B def = hB; [; ?; s ; ck i 2H;k2! is in CdfH and the relativization of B with the set V is a homomorphism from B to A. This gives that A 2 HCdfnmH . We need to show that IGwdfH = IGwdfHcm. The IGwdfHnm IGwdfHcm part is obvious so it is left to check that IGwdfHnm IGwdfHcm. Let A be an algebra from GwdfHnm. The unit element V of A is a normal Gws! {unit therefore it is a disjoint union of weak spaces. Denote the set of weak spaces building V by . We put U to be the union of the bases of elements of . We de ne an accessibility relation R on as: for any 1 ; 2 2 1 R2 i (9q 2 1 )(9 2 H ) q 2 2 : 31
It follows that 1 R2 ) base(1 ) = base(2 ). (Here we used that V is a normal Gws! {unit.) Let R be the transitive, symmetric, re exive closure of R. Clearly, R is an equivalence relation. We have that 1 R 2 ) base(1 ) = base(2 ). For each 2 =R we put U to be base() for some 2 , and we x a function f : U ! U that is the identity on U . Now we can map the sequences of V into subsets of ! U in a following way. For any q 2 2 2 =R
m(q) def = fs 2 ! U j f s = q; s qg: Claim 3.19 The statements below hold. For all q; q0; s 2 V (i). m(q) and m(q0 ) are disjoint if q and q0 are distinct. (ii). s 2 m(q) ) s[i=u] 2 m(q[i=f (u)]) if u 2 U . (iii). s 2 m(q) ) s 2 m(q ) if 2 H . Proof (i): Suppose that s 2 m(q) \ m(q0). Then q s q0 so q and q0 are in the same weak space of V , say in . Since 2 for some 2 =R we infer that q = f s = q0 . (ii): Suppose that s 2 m(q), q 2 2 2 =R. Since q q[i=f (u)] we have q[i=f (u)] 2 2 . Now f s[i=u] = q[i=f (u)], s[i=u] s q q[i=f (u)]
cll1
shows that (ii) holds. (iii): Suppose that s 2 m(q), q 2 2 2 =R. If 0 is the weak space of q then R0 and so R 0 , 0 2 . We have s q, f s = q and therefore f s = q and s q (we used that is bounded). We put V =
[
q2V
m(q). (ii) of the previous claim shows that V is a com-
pressed Cartesian space, (iii) gives that s (V ) = V whenever 2 H .
Claim 3.20 The statements below hold. [ m(q0 ). (i). fs 2 V j 9u 2 U s[i=u] 2 m(q)g = (ii). fs 2 V j s 2 m(q)g =
q0 2ci fqg
[ q0 2s fqg
m(q0 ). 3:19(ii)
Proof [ (i): 0 If s[i=u] 2 m(q) then s = s[i=u][i=s(0 i)] 2 m(q0 [i=f (s(i))]) m(q ). On the other hand, if s 2 m(q ) for some q 2 ci fqg then q0 2ci fqg
3:19(ii)
s[i=q(i)] 2 m(q0 [i=f (q(i))]) = m(q) since f (q(i)) = q(i); q0 [i=q(i)] = q. (ii): If s 2 V ; s 2 m(q) then s 2 m(q0 ) for some q0 2 V . By Claim 3.19 (iii), s 2 m(q ). Claim 3.19 (i) implies m(q) \ m(q0 ) 6= ; ) q0 = q. 32
cll2
So q0 2 s fqg. On the other hand, if s 2 m(q0 ) for some q0 2 s fqg then s 2 m(q0 ) = m(q). We arrived to the point when the algebra in which A is embeddable and the representation function rep can be de ned: A0 = P (V ) 2 GwdfHcm; for any a 2 A; rep(a) =
[
q2a
m(q):
Clearly, rep is a Boolean embedding of A into A0 since m(q) is not empty (q 2 m(q)), and m(q) and m(q0 ) are disjoint when q and q0 are distinct (see Claim 3.19(i)). To see that[ rep preserves we compute: [ [the extra Boolean [ operations [ m(q0 ) = ci rep(a) = ci m(q) = ci m(q) 3:20( m(q0 ) = = i) q2a
q2a
q2a q0 2ci fqg
q0 2ci a
q2a
q2a
q2a q0 2s fqg
q0 2s a
rep(ci a). [ [ [ [ [ m(q0 ) = m(q0 ) = s rep(a) = s m(q) = s m(q) 3:20( = ii) rep(s a).
Before proving Theorem 2.11 (iii) we state an important result.
Lemma 3.21 Let H be a set of quasi-bounded transformations, A be a GwdfH , 1 and 2 be weak spaces of A. If (9q 2 1 )(9 2 H ) q 2 2 , then base(1 ) base(2 ).
Proof Suppose that u 2 base(1 ), q 2 1, 2 H , j?1((k))j < ! (there is a proper k 2 ! while is quasi-bounded), q 2 2 . Since 1 is a weak space, q[(k)=u] 2 1 . But then fi 2 ! j q[(k)=u] (i) 6= q (i)g ?1 ((k)) is nite so q[(k)=u] q 2 2 . The sequence q[(k)=u] of 2 takes the value u at k so u 2 base(2 ). Proof of Theorem 2.11 (iii) We need tonmshow that under the conditions formulated in the theorem GwdfH IGwdfH . Let A be an algebra from GwdfH . As we did in the proof of (ii) we put V to be the unit element of A, to be the set of weak spaces of A, R to be the accessibility relation on , R to be the equivalence relation generated by R. We show that R0 ) base() = base(0 ) (and so R 0 ) base() = base(0 )). Suppose that R0 . Then (9q 2 )(90 2 H ) q 0 2 0 . We know that 0 has a quasi-inverse in [H ], say 1 : : : n . For any i 2 n + 1 we put i to be the weak space of q 0 : : : i . Lemma 3.21 implies that base(i ) base(i+1 ) (i 2 n). If we note that 0 1 : : :n Id, and so = n then we have base() base(0 ) base(1 ) : : : base(n ) = base(). 33
l-qb
Now we can de ne the set U to be base() for some 2 2 =R. Surely, the de nition is independent from the choice of . We put f : U ! U f g to be the natural embedding (f (u) = hu; i), and : V ! =R to be the function that associates the R class of the weak space of q to a q 2 V . We de ne for any a 2 A; rep(a) = ff (q) q j q 2 ag A0 = frep(a) j a 2 Ag 0 A = hA0 ; [; ?; s ; ck i 2H;k2! It is straightforward to verify that A0 2 GwdfHnm and rep is an isomorphism between A and A0 .
3.4 Quasi-axiomatizing ICdfH
To nd a proof to Theorem 2.15, we examine the situation described in Example 2.13. The key idea is that the presence of a constant sequence can be captured by a quasi-equation. Among those GwdfG 's that have constant sequences in their universes, the simplest in structure are the ones with unit elements of the form ! U (q) for some set U and constant sequence q. A unit element of this form will be called horizontal. (It is easy to see that s (! U (q) ) = ! U (q) .) Any algebra from GwdfH that has a constant sequence q in it's universe can be homomorphically mapped into a GwdfH with horizontal unit element. (Indeed, the relativization with ! U (q) | the function that maps the element x of the algebra into x \ ! U (q) | will be a satisfactory homomorphism.) So any GwdfH that is isomorphic to a CdfH has a homomorphism into some GwdfH with horizontal unit element. The next theorem states that this requirement is not only necessary but also sucient to force a GwdfH to be isomorphic to a H {cylindric set algebra.
Theorem 3.22 Let H be an arbitrary subset of ! ! with the property that for any 2 H , is bounded and has a right quasi-inverse in ! !. Then any algebra A 2 GwdfH , that has a homomorphism into some GwdfH with horizontal unit
th-csg
element, is isomorphic to a H {cylindric set algebra. That is,
(9 set U )(9u0 2 U )(9h : A ! P (! U (hu0 ;u0 ;:::i) )) ) A 2 ICdfH : Before the proof of the Theorem above we present an important lemma.
Lemma 3.23 Suppose U is a set, u0 2 U . There is a cardinal for that if P (W 0 ) is a full compressed H {cylindric set algebra with base U 0 of cardinality bigger than then there is a homomorphism h0 : P (! U (hu0 ;u0 ;:::i) ) ! P (W 0 ). Proof Let be a cardinal bigger than !, the cardinality of U and that of H . 34
lm-d->s
For the algebra P (W 0 ), we de ne the set of weak spaces, the accessibility relations R and R on as we de ned in the proof of Theorem 2.11 (ii). For each weak space 2 we x a sequence q in . We put U 0 ( 2 =R) to be the set fq (i) j 2 ; i 2 !g.
Claim 3.24 The cardinality of jU 0 j + jU j is less than . Proof For a given 1 2 and 2 H there is exactly one 2 2 with 1 R2 . This can veri ed by noting that from q 2 2 , q 2 1 we can infer q0 2 2 when q0 2 1 ( is bounded so q q0 ) q q0 and q 2 2 \ 02 ) 2 = 02 ). For any 2 H , is right quasi-invertible in ! ! so there is a transformation f 2 ! ! with f Id. Suppose for contradiction that there is a k 2 ! for that jf?1 (k)j = !. Then the transformation f takes the value (k) on any coordinate of the in nite set f?1 (k). But this is impossible since f takes the value i on coordinate i for all but nitely many i 2 !. This reasoning shows that f is bounded. For a given 2 2 and 2 H there is exactly one 1 2 with 1 R2 . Suppose that 1 ; 01 2 , q 2 1 , q0 2 01 and q 2 2 3 q0 . Then q q0 and q f q0 f since f is bounded. But this implies that q q f q0 f q0 , q 2 1 \ 01 so 1 = 01 . For a given 2 , jf0 2 j R0 or 0 Rgj 2 jH j. For a given 2 , we have that 0 2 and sit in the same R class if and only if there is a nite series 0 ; : : : n of weak spaces with 0 = , n = 0 and i 2 n ) (i Ri+1 or i+1 Ri ). Therefore, the size of a R class is less than 1 + 2 jH j + (2 jH j)2 + : : : ! jH j. Finally, jU 0 j ! j j ! ! jH j = ! jH j, jU 0 j + jU j < . Let f ( 2 =R) be any function mapping U 0 onto U with the property (8i 2 !)(8 2 ) f (q (i)) = u0 : Such an f exists since not more than jU 0 j-many elements of U 0 are to be mapped on u0 so there are enough elements left to cover the set U ? fu0 g. With the help of f , for each sequence q of ! U (hu0 ;u0 ;:::i) a subset of W 0 can be associated.
m(q) = fs 2 W 0 j f s = q; s 2 2 2 =Rg: As it was presented in the more[complicated situation in the proof of Theorem 2.11 (ii) the function h0 (a) = m(q) is a homomorphism. (Claims 3.19 and q2a
3.20 hold here as well since the original proofs can be adapted to the present situation. ) 35
cl-card
Proof of Theorem 3.22 In Theorem 2.11 (ii) and (iii) IGwdfH = IGwdfHcm was proved under the conditions that for any 2 H , is quasi-bounded and it has a quasi-inverse in [H ]. This holds in our situation so we may assume that A 2 GwdfHcm. The unit element of A is V ! U for some set U . Let U 0 be
a set with cardinality bigger than the cardinality of U and the cardinal that corresponds to the set U according to Lemma 3.23. Let f be a function mapping U 0 onto U . With the help of f , for each sequence q of ! U a subset of ! U 0 can be associated. Let V 0 =
[
m(q) = fs 2 ! U 0 j f s = qg:
m(q). As it was presented in the proof of Theorem 2.11 (ii) the [ function r1 : A ! P (V 0 ); r1 (a) = m(q) is an embedding. q2V
q2a
Let W 0 = ! U 0 n V 0 . Just like V 0 , W 0 is a Gws! {unit and s (W 0 ) = W 0 for any 2 H . So W 0 can be a unit element of a GwdfHcm . According to Lemma 3.23 there is a homomorphism h0 mapping P (! U (hu0 ;u0 ;:::i) ) into P (W 0 ). Let r2 be the composition of the homomorphisms h0 and h (r2 = h0 h) mapping A into P (W 0 ). Since P (! U 0 ) = P (V 0 ) P (W 0 ) the function rep de ned as rep(a) = r1 (a) [ r2 (a) is an embedding and it demonstrates that A 2 ICdfH . In the second part of this subsection we show that if an algebra in GwdfH satis es Qx then it has a homomorphism to GwdfH with horizontal unit element. We assume that H satis es the assumption stated in Theorem 2.19.
Lemma 3.25 If A 2 GwdfH and A j= Qx then there is an ultra lter U of A with the property that ?c(?) x 2 U holds whenever x \ s x = 0 for some ? ! ! 2 [H ] with d? = Idd?. ^
Claim 3.26 8; 1; : : : ; n 2 [H ] 90 ; 10 ; : : : ; n0 2 [H ] : 0 i = i0 . in Furthermore, if d? = 1 d? = = n d? = Idd? then the same holds to 0 ; 10 ; : : : ; n0 (? ! !). Proof We prove by induction. If n = 0 then 0 = . Suppose that the statement holds for n ? 1. Let 00 ; 100 ; : : : ; n00?1 be associated to ; 1 ; : : : ; n?1 . Let and n be the left quasi-inverse of 00 n and respectively. Since [H ] contains all nite 9 transformations we can assume that 00 n (k) = 8 = < k 2? 0 0 00 0 k 2 m ? ? 0 ; = n (k) for some m0 2 !. Setting = , : k m0 k i0 = i00 (i = 1; : : : ; n) gives the required result. 36
ll1
clll1
Proof of Lemma \3.25 Let B = f?c(?)x : ? ! !; 9K ! [H ] ( 2 K ! d? = Idd?); s x = 0g. Qx states that B has the nite intersection 2K property.T We put B to be B 's lter closure, that is, B = fy 2 A : 9X ! B y X g. Claim 3.27 B is closed under s for all 2 [H ]. \ Proof Suppose ?c(?)x 2 B, s x = 0 for some ? ! !; K ! [H ], 2K 2 K ! d? = Idd?. Let ?0 = fi 2 ! ? ? : (i) 2 ?; 8j 2 i ? ? ( (i) = 6 (j ))g. 0 0 is bounded so ? is nite. If ? = f 1 ; : : : ; n g then let 0 = [ 1 ; ( 1 )] : : : [ n ; ( n )]. Let ?00 = ?0 [ (? ?f ( ) : 2 ?0 g), 1 = (0 )[?=Id]. Now we have that 0 1 d! ? ? = d! ? ? and for all i 2 ? 1?1 (i) = fig. Using Claim 3.26, 0 0 we pick\corresponding and ( 2 K ). Then 0 = \ transformations \ 1 and to \ s0 s10 s x = s0 s10 s x = s0 s 0 s1 x = s0 00 s0 s1 x. Not 2K
2K
2K
2K
ing that 0 0 0d?00 = Idd?00 we have ?c(?00 ) s0 s1 x 2 B . So ?c(?00 ) s0 s1 x =6 ?s0 c(?) s1 x =6 ?s0 s1 c(?) x =5 ?s c(?) x = s (?c(?) x) gives s (?c(?) x) 2 B . Finally, suppose that x 2 B . Then there are x1 ; : : : ; xn 2 B with x n \
n \
n \ 1
xi .
So s x s xi = s xi 2 B giving s x 2 B . 1 1 Let U be a maximal lter extending B with the property that it is closed under s for all 2 [H ]. We can use Zorn's lemma to construct such a lter. We show that U is indeed an ultra lter. Suppose not. Then there is a y such that neither y nor ?y\is in U . y cannot be added to \ U so there are \ z 2 U , K ! [H ] such that z \ s y = 0. Then 0 = z \ s y s (z \ y) giving 2K
2K
2K
?(y \ z ) 2 B . Similarly, ?(?y \ z 0 ) 2 B . Therefore U 3 ?(?y \ z 0 ) \?(y \ z ) = ?((y [ z 0 ) \ (z [ z 0) \ (z [ ?y)). But all the three terms in the last expression are in U so their intersection is in U giving that U contains an element and its complement.
Our algebra A 2 GwdfH is a normal Boolean algebra with operators. (Cf. e.g. [HMTI] De nition 2.7.1.) We can de ne its canonical embedding algebra or ultra lter extension A as follows (see De nition 2.7.4. in [HMTI]). Let V be the set of ultra lters of A. If O is any of the operators fs : 2 H g[fci : i 2 !g and X V then we de ne OX = fU 2 V : (9S 2 X )(8x 2 U ) Ox 2 S g: Then A is hP (V ); [; ?; s ; ci i 2H;i2! . 37
clll2
As stated in Corollary 3.5, IGwdfHnm is axiomatized by (*1){(*7) (de ned in De nition 3.3) together with the de nition of ci (i 2 !). All that axioms but ci (?ci x) = ?ci x and s (?x) = ?s x are positive, that is, they do not contain any occurance of the symbol ? for complementation. But the two exceptions are equivalent (in the presence of the others) with some positive formulas:
ci (?cix) = ?ci x i ci 0 = 0; ci (x ^ ci y) = cix ^ ciy s (?x) = ?s x i s (x ^ y) = s x ^ s y: Since positive equations valid in a Boolean algebra with operators still hold in the ultra lter extension of the algebra, (see e.g. [HMTI] Remark 2.7.15) we can conclude that the ultra lter extension A of A 2 GwdfHnm is in GwdfHnm. Furthermore, under the hypothesis of Theorem 2.19, IGwdfH = IGwdfHnm holds, so the conclusion stated above holds to the class GwdfH as well. Finally, we note that every Boolean algebra with operators is embeddable into its ultra lter extension. (The map that associates fU 2 V : x 2 U g to an element x in A will suce.)
Lemma 3.28 If A 2 GwdfH and A is its ultra lter extension then the atom a = fU g has the property x \ c(?) a s x (? ! !; 2 [H ]; d? = Idd?; x 2 A ) provided that U is an ultra lter of A that satis es the property stated in Lemma
ll2
3.25.
Proof It is enough to show that the condition x \ c(?)a s x holds for all atoms of A . Suppose x = fF g. If F 62 c(?) a then there is nothing to show. If F 2 c(?) a then 8y 2 F c(?) y 2 U . Suppose for contradiction that x 6 s x, that is, 9y 2 F s y 62 F . Then ?s y 2 F , w = y \ ?s y 2 F . We can conclude that c(?) w 2 U . On the other hand, w \ s w = 0 gives ?c(?)w 2 U and that is a contradiction.
Lemma 3.29 If A 2 GwdfHcm and A has an atom a with x \ c(?)a s x whenever ? ! !; 2 [H ]; d? = Idd?; x 2 A then A has a homomorphism
ll3
to a GwdfH with horizontal unit element.
Proof We denote the base of A by U 0 and the [ universe of A by A . Let q be a sequence in a, V = fq : 2 [H ]g, V = c(m)V . Put U0 to be the m2! range of q. Fix an arbitrary element u 2 U0 . Denote U 0 ? U0 [ fug by U . Let f be the function that is the identity on U 0 ? U0 and the constant u on U0 . Claim 3.30 x 2 A; s; s0 2 V ; s 2 x; f s = f s0 ! s0 2 x. Proof Let ? = fi 2 ! : s(i) 62 U0g. Since s q for some 2 [H ], ? is nite. Clearly, ? = fi 2 ! : s0 (i) 62 U0 g so sd? = s0 d?. We have that s0 q 0 for some 38
clll3
0 2 [H ]. s = q [?=s], s0 = q 0 [?=s]. Let and 0 be the right quasi-inverses of and 0 . Since all the nite transformations are in [H ], we can choose and 0 that they satisfy the following additional properties: d? = 0 d? = Idd?, d! ? ? = 0 0 d! ? ?. Then s = (q [?=s]) = q [?=s] = q 0 0 [?=s] = (q 0 [?=s]) 0 = s0 0 . Since s 2 x \ c(?) a s x, we get that s0 0 = s 2 x. (We used that a s a and so q 2 a.) If we suppose for contradiction that s0 62 x then s0 2 ?x \ c(?) a s 0 ? x and so s0 0 2 ?x but that cannot be the case.
Now we can de ne a homomorphism that maps A into P (! U (hu;u;:::i) ): h(x) = ff s : s 2 x \ V g: It is straightforward to verify that h preserves \, while Claim 3.30 proves that it preserves complementation. The following two claims prove that h preserves the extra Boolean operations. Claim 3.31 cih(x) = h(ci x). Proof r 2 ci h(x) ) r = (f s)[i=r(i)] for some s 2 x. But r = (f s)[i=r(i)] = f (s[i=r(i)]) 2 h(ci x) since s[i=r(i)] 2 ci x. On the other hand, r 2 h(ci x) ) r = f s for some s 2 ci x. So there is s0 2 x with s0 [i=s(i)] = s. But r = f s = f (s0 [i=s(i)]) = (f s0 )[i=f (s(i))]) 2 ci h(x)
Claim 3.32 s h(x) = h(s x). Proof r 2 s h(x) , r 2 h(x) , 9s0 2 x f s0 = r . r = f s for some s 2 V so f s0 = r = f s . Using Claim 3.30 we have that s0 2 x , s 2 x. But s 2 x , s 2 s x , r = f s 2 h(s x) completes the proof.
Summarizing the results of this subsection, we prove Theorem 2.19.
Proof of Theorem 2.19 Suppose that the algebra A is a model of Qx [ Eq(IGwdfH ). Lemmas 3.25 and 3.28 state that the ultra lter extension A of A has an atom with certain properties. Since the ultra lter extension satis es all the positive equations valid in the algebra, we have that A j= Eq(IGwdfH ). Using Theorem 2.11 we conclude that A is isomorphic to an algebra B 2 GwdfHcm. Lemma 3.29 implies that B has a homomorphism to a GwdfH with horizontal unit element. Since A is embeddable into its ultra lter extension and, in that way, into B, A also has a homomorphism to a GwdfH with horizontal unit element. Finally, Theorem 3.22 states that the above mentioned result is sucient to show that A 2 ICdfH . 39
clll4
clll5
3.5 Theorem 2.30: the cylindric reduct of Mod( 1 2) H
;H
The proof will have the following structure. Any model of H1 is actually an extension of a model of EqGwdfHnm. But Theorem 2.11 (i) imply that a model of EqGwdfHnm is isomorphic to a GwdfHnm We will examine the extra abstract constant d01 . (In the presence of the substitutions the other dij 's are term de nable so it is enough to concentrate on d01 .) Finally, the proper behaviour of d01 will lead us to a representation function that proves our theorem. Recall from Notation 2.8 that if q1 are q2 sequences then q1 q2 indicates that they agree on all but nitely many coordinates.
Notation 3.33 Let H be a set of transformations of !. Suppose that = h1 : : : n i 2 H . We introduce the notation s x for the term s1 : : : sn x. Remark 3.34 Let H be a set of transformations of ! with the property that [H ] is rich. (Recall that [H ] is the subsemigroup of h! !; i generated by H [f[i=j ] j i; j 2 !g.)
nt-stau
rm-H4
In the de nition of a rich semigroup the existence of two special elements s^ and p^ were stated. We x two elements of [H ] with the required property and denote them by s^ and p^. We also x ss^ and sp^ to stand for the terms s 1 and s 2 respectively where 1 = s^, 2 = p^, and 1 ; 2 2 H . It is provable from H1 that for any choice of 1 2 H with 1 = s^, s 1 is the same operation of the algebra so ss^ (and similarly sp^) is well de ned. s^(i) 6= s^(j ) if i 6= j since p^ s^ = Id. There is a permutation f of ! that takes s^(0) to 0 and s^(1) to 1 and xes all the elements of ! ? f0; 1; s^(0); s^(1)g. (f is one of [0; s^(0)] [1; s^(1)], [1; s^(1)] [0; s^(0)] or [0; 1] depending on whether any of s^(0) = 0, s^(0) = 1, s^(1) = 0 or s^(1) = 1 hold.) Since f is nite f 2 [H ]. So there is a H 2 H for that H = f . We formalize a set of equations that are all consequences of H1,H2.
s[0=1]x s[1=0]x ci d01 s[0;1]d01 s H ss^d01
c0(d01 ^ x); c1(d01 ^ x); d01 whenever i 2; (H4) d01; d01: Here s[i;j] x stands for the term s x where 2 H and = [i; j ]. Since [H ] is = = = = =
rich, all the nite transformations (including [i; j ]) belong to [H ].
Lemma 3.35 Let H be a set of transformations of ! with the property that [H ] is rich. Suppose that A is a GwdfH extended with the abstract18 constant d01
18 Here the adjective \abstract" means that this constant is NOT necessarily the set theoretically de nable diagonal element, but any constant that satis es the required equations. It will be shown that this \abstract" constant need to have a nice structure similar to that of the set theoretic diagonal element.
40
lm-2
that satis es H4. If x is an element of A and hu0 ; u1 ; u2 ; : : :i is a sequence from the universe of A then hu0 ; u1 ; u2 ; : : :i 2 d01 ^x implies hu0 ; u0; u2 ; : : :i 2 x;
hu1 ; u0; u2 ; : : :i 2 x; hu1 ; u1; u2 ; : : :i 2 x:
Proof Suppose that hu0; u1; u2; : : :i satis es the premise of the lemma. Then hu0 ; u1 ; u2; : : :i 2 c0 (d01 ^ x) H=4 s[0=1] x. This implies hu0 ; u1 ; u2 ; : : :i [0=1] = hu1 ; u1 ; u2; : : :i 2 x. Using s[1=0] x H=4 c1 (d01 ^ x) we can get hu0 ; u0 ; u2; : : :i 2 x. s[0;1]d01 H=4 d01 implies hu1; u0; u2; : : :i 2 d01 . If we suppose for contradiction that hu1 ; u0; u2 ; : : :i 62 x then hu1 ; u0 ; u2 ; : : :i 2 ?x and according to the previous observation we can conclude that hu1 ; u1 ; u2; : : :i 2 ?x and that is impossible. Lemma 3.36 Let H; A and d01 be as in Lemma 3.35. If u and v are the rst two elements of a sequence q of d01 then any sequence r of an arbitrary x 2 A can be modi ed in any of its positions from u to v provided that q and r are in the same weak space. In other words, if q 2 d01 , r 2 x for some x 2 A, q r, r(k) = q(0) = u for some k 2 ! and q(1) = v then r[k=v] 2 x. Proof First suppose that k =6 1. Let q = (r [0; k])[1=v]. Clearly, q is in the weak space of q (and of r). Therefore, q 2 ci1 : : : cin d01 where ? = fi1; : : : in g is the set where q and q do not agree (it is nite). Now q (0) = (r [0; k])(0) = r(k) = u and q (1) = v so 0; 1 62 ? and ci d01 H=4 d01 whenever i 2. We can conclude that q 2 d01 . We put z to be an element of ! ? Rng(^s). [H ] is rich so ! ? Rng(^s) is not empty. (q p^)[z=r(1)] s^ = q 2 d01 implies that (q p^)[z=r(1)] 2 ss^d01 . (q p^)[z=v][^s(1)=r(1)] s^ = r [0; k] 2 s[0;k] x, (q p^)[z=v][^s(1)=r(1)] 2 ss^s[0;k] x. Furthermore, (q p^)[z=r(1)] = (q p^)[z=v][^s(1)=r(1)] [z; s^(1)]. We put r def = (q p^)[z=r(1)] H?1 . Summarizing all the results we get that r 2 s H ss^d01 H=4 d01 and r 2 s H s[z;s^(1)]ss^s[0;k]x, r (0) = q(0) = u, r (1) = q(1) = v. Using Lemma 3.35 we conclude that r [0=v] 2 s H s[z;s^(1)] ss^s[0;k] x. This means that r [0=v] H [z; s^(1)] s^ [0; k] 2 x. Then r [0=v] H [z; s^(1)] s^ [0; k] = ((q p^)[z=r(1)] H?1 H )[^s(0)=v] [z; s^(1)] s^ [0; k] = (q p^ [z; s^(1)])[^s(1)=r(1)][^s(0)=v] s^ [0; k] = (q p^ s^)[1=r(1)][0=v] [0; k] = ((r [0; k])[1=v][1=r(1)] [0; k])[k=v] = (r [0; k] [0; k])[k=v] = r[k=v] 2 x. If k = 1 then r 2 x ) r [1; 2] 2 s[1;2] x ) (r [1; 2])[2=v] 2 s[1;2] x ) (r [1; 2])[2=v] [1; 2] = r[1=v] 2 x. Proof of Theorem 2.27 (1) Surely, any representable cylindric set algebra is a subalgebra of a reduct of a model of H1,H2. 41
lm-3
To see the other direction, suppose that A0 is a model of H1,H2. A0 j= H 1 means that all the valid equations of GwdfHnm hold in A0 . Naturally, some operations of tH are term functions in GwsH . Since IGwdfHnm is a variety, the tH {reduct of A0 is isomorphic to a GwdfHnm , say to A. We will use only that A 2 IGwdfH . The image of d01 under the isomorphism is an element of A not necessary the set theoretic (0,1){diagonal. The image of any other diagonal constant can be computed from the image of d01 since dij is term de nable from d01 in GwdfH . This shows that to represent A0 , it is enough to represent A that is to nd a isomorphism rep mapping the cylindric reduct of A into a RCA! . The universe V of A is a Gws! {unit therefore it is a (disjoint) union of weak spaces. Denote the set of weak spaces building A by . For any weak space from , base() is the base set of . d01 de nes an equivalence relation on base() in a following way: for any u; u0 2 base()
u u0 i (9q 2 d01 \ ) q(0) = u; q(1) = u0
Claim 3.37 is re exive. Proof Suppose u 2 base(). Let hu0; u1; u2; : : :i 2 arbitrary. It is obvious that hu; u; u2; : : :i 2 . Now hu; u; u2; : : :i 2 V = c1 d01 . So (9v 2 base()) hu; v; u2 ; : : :i 2 d01 . Applying Lemma 3.35 to the case when x = d01 it gives that hu; u; u2; : : :i 2 d01 Claim 3.38 is symmetric. Proof Follows from Lemma 3.35 when applied to the case x = d01. Claim 3.39 is transitive. Proof Follows from Lemma 3.36 when applied to the case x = d01, k = 0. Our aim is to embed the cylindric reduct of the not fully represented algebra
A into a Gws! . For saving energy and not losing what we have already achieved,
we are looking for a representation function rep that acts not only on the algebra level but deeper, on the sequence level hiding in the elements of A. In other words, rep(x) = ff (q) j q 2 xg where f if the sequence level function mapping V into the universe V of some algebra A . To ensure that f (d01 ) is the set theoretic constant we want for any q; q0 2 2 that (8i 2 !) q(i) q0 (i) should imply f (q) = f (q0 ). This observation gives us a hint for an adequate de nition of f . 42
cl-1
cl-2
cl-3
For every 2 , we de ne U to be the set (base()= ) fg. U corresponds to the classes of base(). Clearly, if and 0 are distinct then U and U0 are disjoint. For every q 2 , we de ne
f (q) = hhq(0)= ; i; hq(1)= ; i; hq(2)= ; i; : : :i The f image of a 2 is also a weak space in ! U . Let it be denoted by . Now we can de ne the image algebra A 2 Gws! :
V =
[ ;
2
A = frep(x) j x 2 Ag; A = hA ; \; ?; ci ; dij i: One can easily verify that the above de ned rep is an isomorphism between the cylindric reduct of A and a generalised weak cylindric set algebra. Finally, the result IGws! IGs! = RCA! published in [HMTII] Theorem 3.1.103. completes the proof.
3.6 Further observations on Mod( 1 2) H
;H
We have seen that a model A0 of H1,H2 is isomorphic to an algebra A whose tH reduct is a GwdfH . In the previous proof another isomorphism rep was de ned that mapped the cylindric reduct of A into A 2 Gws! . A can be expanded with the abstract operations s ( 2 H ) as follows
s y def = rep s rep?1 y: The expanded algebra is denoted by A as well.
We examine the behaviour of the new, abstract operations. In this subsection we collect plenty of the characteristic features that will be useful in the proof of Theorem 2.30 (3). All the notations (A; A ; ; ; f; rep) that were introduced in the previous subsection are kept. The set of weak spaces of A is denoted by . and is in a one-to-one correspondence in a natural way. We have that base( ) = (base()= ) fg. >From now we suppose that all the elements of H are bounded transformations of !.
Claim 3.40 If 2 H (and so is bounded) and 1 ; 2 2 then the statements below hold.
s 1 2 s 1 2
or i
s 1 \ 2 = ;. (9q 2 2 )(q 2 1 ). 43
cl-bound
(82 2 )(9!1 2 )(s 1 2 ). (Here 9! stands for `there exists exactly one'.)
Proof Suppose that q 2 s 1 \ 2 . For all q0 2 2, q0 2 1 since q 2 1 and fi 2 ! j q (i) 6= q0 (i)g ?1 fi 2 ! j q(i) 6= q0 (i)g is nite. So 2 s 1 . To prove the last statement suppose that q 2 2 2 . Put 1 to be the weak space of q . Clearly, q 2 1 and so 2 s 1 . If 0 2 and 2 s 0 then s 1 \ s 0 = s (1 \ 0 ) 2 shows that 1 \ 0 6= 0 and so 1 = 0 . That claim shows that the following de nition is explicit.
De nition 3.41 For each 2 H , we de ne a transformation ^ of : ^ (2 ) = 1 if 2 s 1 :
d-sonL
Since and are in one-to-one correspondence, ^ can be viewed as a transformation of : 1 ; 2 2 , ^ (2 ) = 1 if ^ (2 ) = 1 . We present an important property of these transformations.
Claim 3.42 Suppose that 1 ; : : : ; n and 10 ; : : : ; m0 are elements of H . If n : : : 1 m0 : : : 10 then ^1 : : : ^n and ^10 : : : ^m0 are the same
cl-hatjo
transformations of (and of as well).
Proof Suppose that 2 . ^1 : : : ^n () = 0 if sn : : : s1 0. Then for any q 2 , q n : : : 1 2 0 . Since n : : : 1 m0 : : : 10 and they are bounded, q n : : : 1 q m0 : : : 10 . Then q m0 : : : 10 2 0 , sm0 : : : s10 0 , ^10 : : : ^m0 () = 0 and this was to prove. Remark 3.43 The domain of the function^de ned in De nition 3.41 is H . The previous claim implies that^can be de ned on [H ] as well. If 2 [H ] ! ! then we set ^ def = ^1 : : : ^n where n : : : 1 . Clearly, ^ does not depend on the choice of 1 ; : : : ; n . The new^: [H ] ! has the following properties: If ; 2 [H ]; then ^ = ^. If ; 2 [H ] then d = ^ ^. Claim 3.44 When ^(2 ) = 1 for some 2 H; 1 ; 2 2 then base(2 ) base(1 ), 2 = 1 \ 2 base(2 ). 44
r-ph
cl-base
Proof Suppose that 2 s 1 , q 2 2. For all u 2 base(2 ), q[(0)=u] 2 2 and so q[(0)=u] 2 1 . But (q[(0)=u] )(0) = u 2 base(1 ). It shows that base(2 ) base(1 ). To prove the second statement suppose that u; u0 2 base(2 ), u 2 u0 . There is a q2 2 d01 \ 2 for that q2 (0) = u, q2 (1) = u0 . We x a k 2 ! for that (0) = 6 (k) and a nite transformation h of ! that maps (0) to 0 and (k) to 1. Since [H ] is rich h 2 [H ] and so there is 2 H for that = h. We put = q2 h [1; k]. q2 2 d01 and s s s[1;k] d01 = dh(([1;k](0)));h(([1;k](1))) = q1 def d01 implies that q1 2 d01. q2 2 2 and 2 s 1 gives that q1 2 1. Finally, q1 (0) = u, q1 (1) = u0 shows that u 1 u0 . For the other direction suppose that u; u0 2 base(2 ), u 1 u0 . There is a q 2 1 \ d01 for that q(0) = u, q(1) = u0. Pick any q2 2 2 with q2 (0) = u, = q2 h [1; k]. As in the previous case q1 2 1 . q2 (1) = u0 . We put q1 def q q1 so they disagree on a nite set ? = fi1 ; : : : ; in g. 0; 1 62 ?. We conclude that q1 2 ci1 : : : cin d01 = d01 . Using again s s s[1;k] d01 = d01 and q1 2 d01 we get that q2 2 d01 . This proves that u 2 u0 . De nition 3.45 We de ne the function [ [ b: fh; ig base( ) ! base( ) 2 H; 2
d-em
2
as follows. b; (h ; i) = h 0 ; ^ ()i if 0 2 base(^())=^ () and 0 holds. The previous claim proved that b is well de ned. We collect some properties of the function b de ned above. Claim 3.46 (1). q 2 2 ) b; q 2 V . (2). q; q0 2 2 ) b; q b; q0 . (3). b; is injective. (4). (8 2 )(9!0 2 )(8q 2 ) b; q 2 0 . Proof (1): Since q 2 V , there is a s 2 V for that f (s) = q. The weak space of s in A is , and that of q in A is . We show that b; q = f (s ) 2 V by checking that they take the same value on an arbitrary i 2 !. (b; q )(i) = b; (q((i))) = b; (f (s)((i))) = b; (hs((i))= ; i) = hs((i))=^ () ; ^ ()i = f (s )(i) To prove (2) we need only that is bounded. (3) follows from the de nition, and (4) from (2).
45
cl-propb
Remark 3.47 If a function g satis es condition (4) of Claim 3.46 then for each 2 H , a transformation ^g of can be de ned: ^g () = 0 i (8q 2 ) g; q 2 0 : When g coincides with b then we get back ^ , that is ^b = ^ . Claim 3.48 Let 2 , 1 ; : : : ; n and 10 ; : : : ; m0 be elements of H . (5). If ^1 : : :^n ( ) = then b1 ;^2 (:::^n ( ):::) b2 ;^3 (:::^n ( ):::) : : :bn; is the identity mapping of base( ). (6). If ^1 : : : ^n ( ) = ^10 : : : ^m0 ( ) then b1 ;^2 (:::^n ( ):::) b2 ;^3 (:::^n ( ):::) : : : bn ; = = b10 ;^20 (:::^m0 ( ):::) b20 ;^30 (:::^m ( ):::) : : : bm0 ; Proof First we note that the expression formulated above is correct. The range of bi ;^i+1 (:::^n ():::) is contained in base(^i (: : : ^n () : : :)) and that is the domain of bi+1 ;^i (:::^n ():::) . To check the validity of the expression in (6) we evaluate the two sides of the equation on an arbitrary element of base( ), say an hu= ; i. When the left side of the equation is applied to that element we get hu=(^1 :::^n )() ; (^1 : : : ^n )()i and that is the same point as the image of hu= ; i under the function that appear on the right side of the equation. (5) follows from the de nition.
Notation 3.49 We de ne a new composition operation mapping V H into V : q def = b;[q] q where [q] is the weak space of q in A . Now a direct de nition can be given to s ( 2 H ). Claim 3.50 For all 2 H; y 2 A s y = fq 2 V j q 2 yg Proof As a side result, in the proof of Claim 3.46 (1) we showed that if s 2 V; q 2 V ; f (s) = q then f (s ) = q . Now we can compute that q 2 s y , s 2 s rep?1 y , s 2 rep?1 y , f (s ) 2 y , q 2 y. We arrived to the point when a new class of set algebras can be introduced. A new algebra has a deeper structure than a GwsH does and its operations are computable from the structure of the algebra. 46
rm-sigmab
cl-prop5
nt-circ
cl-newd2
De nition 3.51 An algebra A = hA; [; ?; ck ; dkl ; s ik;l2!;2H is called a nonstandard generalised weak H {cylindric set algebra with equality (or brie y a nstGwsH ) if hA; [; ?; ck ; dkl ik;l2! is a generalised weak cylindric set algebra
df-nonst
and there is a function b with the following properties: Let V be the unit element of A. We[put to be the set of weak spaces building V . The domain of b is f; g base(), the range is a subset of
[
2
base().
2H;2
b satis es the properties (1)-(6) listed in Claim 3.46 and Claim 3.48 where
^ = ^b is de ned as in Remark 3.47. We put : V H ! V to be the function q def = b;[q] q . Clearly, the properties of b gives that b;[q] q is an element of V . Then s y = fq 2 V j q 2 yg. If A is a nstGwsH and b is a function with the required properties then b is
called a base transformation function of A. Note that the base transformation function is not uniquely determined. The theorem below is a summary of the results that have achieved so far in this subsection.
Theorem 3.52 Any model of H1,H2 is isomorphic to a nstGwsH . Theorem 3.53 GwsH nstGwsH . Proof Any GwsH is a nstGwsH with the identity as base transformation function. (8 2 H )( 2 )(8u 2 base()) b;(u) def = u. Claim 3.54 Suppose that A 2 nstGwsH . Then Claim 3.42 is true in A. Proof As usual, is the set of weak spaces of A. Pick an arbitrary element of , say , and an arbitrary sequence q in . Using the de nition of ^ in Remark 3.47 we have that ^1 : : : ^n () and ^10 : : : ^m0 () are the weak spaces of b1 ;^2 (:::^n ():::) b2;^3 (:::^n ():::) : : : bn; q n : : : 1 = q n : : : 1 and b10 ;^20 (:::^m0 ():::) b20 ;^30 (:::^m ():::) : : : bm0 ; q m0 : : : 10 = q m0 : : : 10 each respectively. Since n : : : 1 m0 : : : 10 , we have that q n : : : 1 q m0 : : :10 . This fact and Claim 3.48 implies that q n : : :1 q m0 : : :10 and so they are in the same weak space, that is ^1 : : : ^n () = ^10 : : : ^m0 (). 47
th-nonst th-st-nst
cl-hatjo2
Remark 3.55 As we did in Remark 3.43, we can introduce the function ^ : [H ] ! as ^ def = ^1 : : : ^n when 2 [H ]; 1 ; : : : ; n 2 H and n : : : 1 .
r-pb
In the same sense, the domain of b can be expanded
b:
[
2[H ];2
f; g base() !
[
2
base()
= b1 ;^2 (:::^n ():::) b2 ;^3 (:::^n ():::) : : : bn ; b; def where 2 [H ]; 1 ; : : : ; n 2 H; 2 and n : : : 1 . If 10 ; : : : ; m0 2 H and m0 : : : 10 then ^1 : : : ^n = ^ = ^10 : : : ^m0 so ^1 : : : ^n () = ^10 : : : ^m0 (). This gives that (6) of Claim 3.48 is applicable showing that the above de nition of b does not depend on the choice of 1 ; : : : ; n . We collect some properties of our new^functions. ; 2 [H ]; 2 . (1). implies ^ = ^. (2). d = ^ ^. (3). Id implies that b; is the identity mapping of base(). (4). b ; = b;^() b; . (5). ^() = ^() implies b; = b; . (1) and (2) can be found in Remark 3.43, (3) and (5) are only reformulations of (5) and (6) of Claim 3.48, and (4) is a consequence of (6) of Claim 3.48.
Lemma 3.56 Let A be a subdirect irreducible model of H1,H2,H3. Then either ?c0 (?d01 ) = 1 or ?c0 (?d01 ) = 0. Proof Suppose that in a model A of H1,H2,H3 neither ?c0(?d01) = 0 nor c0 (?d01) = 0 holds. We show that the relativizations with ?c0(?d01) and c0 (?d01) are homomorphisms, but not isomorphisms, and A is a subdirect product of the two image algebras. This proves that A cannot be subdirectly
th-subd
irreducible. A relativization with some element R is a homomorphism if and only if R is closed under the operations of the algebra. H3 implies that ?c0 (?d01 ) and c0 (?d01) are closed. In the image algebras ?c0(?d01) = 1 or c0 (?d01) = 1 will hold showing that they cannot be isomorphic to A. Finally, an element of A is solely determined by its two image under the two relativizations.
Lemma 3.57 If ?c0(?d01) = 1 holds in A 2 nstGwsH then A is a GsH . 48
l-extr
Proof Let A be the universe, V the unit element, the set of weak spaces of A. ?c0 (?d01 ) = 1 shows that for all 2 , jbase()j = 1. Introduce the set theoretical operations on A:
~s x def = fq 2 V j q 2 xg: Then A0 def = hA; [ ?; ck ; dkl ; ~s ik;l2!;2H 2 GsH . We show that ~s = s and so A = A0 2 GsH . s x = x since the axiom s (x ? c0(?d01 )) = x ? c0(?d01) in H3 can be reduced to that form using ?c0 (?d01 ) = 1. On the other hand, ~s x = x is a result of that fact that in V all the sequences are of the form hu; u; u; : : :i for some u.
3.7 The witness tree method
In this subsection we build a machinery called the witness tree method to prove Theorem 2.30. The essence of the method is the following. The failure of an equation in a given algebra can be captured by a tree structure that re ects the local nature of the algebra. This tree can be cut out from the algebra and moved to another one with better properties to testify the failure of the same equation therein. In our case we will show that an equation that is not satis ed in a nstGwsH cannot hold in all the GwsH s.
De nition 3.58 Let H be a set of transformations of !. We say that = hV; E; r; U; ; "; li is a witness tree of type H if the following requirements are
ful lled. (i). hV; E i is a directed tree. r 2 V . The edges are directed from the root r towards the leaves. (ii). U; and " are functions with domain V . If v 2 V then Uv is a set, v is a term of type t=H , and "v is either 3 or 63. (Note that we used the less conventional notation Uv instead of U (v).)
(iii). l is a function with domain E . If e = vv~ 0 2 E is an edge then le is one of '=', '[k=u]' or '' where k 2 !, u 2 Uv , 2 H . We will call le the label of the edge e. (iv). If e = vv~ 0 2 E is an edge then Uv Uv0 . Furthermore, if le is not '' for some 2 H then Uv = Uv0 . (v). For every vertex v, v , "v , the number of children of v, the labels of the edges pointing to the children of v, and the value of and " on of the children vertexes of v are in a connection indicated in the following table. 49
d-wt
v , " v (?%); 3 (?%); 63 (%1 [ %2 ); 3 (%1 [ %2 ); 63
ck %; 3 ck %; 63 s %; 3 s %; 63 X; 3 X; 63 dkl ; 3 dkl ; 63
The number of children 1 1 1 2 1
jUv j 1 1 0 0 0 0
The label of The value an edge of , " on pointing to a a child child vertex = %; 36 = %; 3 = %i ; 3 = %i ; 63 [k=u] %; 3 [k=u] %; 63
? ? ? ?
%; 3 %; 63
where i is 1 or 2 for i 2 f1; 2g where u 2 Uv for each u 2 Uv
? ? ? ?
De nition 3.59 Let be a witness tree. We de ne a function : V ! ! ! by recursion. For the root r, r = Id. If e = v1~v2 2 E and v1 is already de ned then v2 is set to v1 when le is '=' or '[k=u]' for some k 2 !; u 2 Uv1 and to v1 when le is ''. De nition 3.60 Let % be a term. Then M% denotes the set of those transformations f of ! for that exist 1 ; : : : ; n 2 H with the property that f = 1 : : : n 6 j ) but and s1 ; : : : ; sn are letters of %. We allow i to be equal with j (i = in that case si and sj refer to the multiple occurrence of the same letter in %. Clearly, n is smaller than or equal to the length of term %. Claim 3.61 The set M% de ned in the previous de nition is nite. Proof The term % has only nite number of letters so there is only nitely many possible choices of 1 ; : : : ; n .
De nition 3.62 Let be a witness tree. Then M denotes M r . Claim 3.63 For each vertex v of , v 2 M . De nition 3.64 M =; M 6= denotes the following subsets of M M . M = def = fh; 0 i j ^ (9 vertex v; v0 )( = v ^ 0 = v0 )g M 6= def = fh; 0 i j 6 0 ^ (9 vertex v; v0 )( = v ^ 0 = v0 )g 50
d-nuv
d-Mr
cl-Mfin
d-M cl-nuM d-M=
Both M = and M 6= are nite.
De nition 3.65 Let be a witness tree. We say the is not degenerated if (z) (8 vertex v1 ; v2 )(8 2 [H ]) v1 v2 ) Uv1 Uv2
d-degen
holds.
De nition 3.66 Let be a witness tree. We say that a function Q is a valu-
d-validq
Claim 3.67 A valuation Q of the vertex set of a witness tree and the value
cl-vq
ation of the vertex set of if the conditions below hold. (1). The domain of Q is the vertex set of . (2). For all vertex v, Qv is an !-sequence over Uv . (3). If v1 and v2 are two adjacent vertexes of then the connection between Qv1 and Qv2 is determined by the label of the edge that joins v1 and v2 . If the label is '=' then Qv2 = Qv1 . If the label is '[k=u]' then Qv2 = Qv1 [k=u]. If the label is '' then Qv2 = Qv1 .
of Q on the root of uniquely determine each other.
Proof If we know the value of Q on the root then we can compute the values of Q on other vertexes going step-by-step towards the leaves using (3) of De nition 3.66. To verify (2), we need only that Uv1 Uv2 whenever v1~v2 is an edge. (See the table in De nition 3.58.)
Claim 3.68 Let be a witness tree with root r, Q be a valuation of the vertex set. Then
(9c1 2 !)(8 vertex v)(8i 2 !) Qv (i) 6= (Qr v )(i) ) i c1 :
Proof To get an insight into the relation of Qv and Qr v , we take an example rst. Suppose that we nd the sequence 1 , [k1 =u1], 2 , [k2 =u2], 3 , 4 , [k3 =u3], 5 of labels on the r ! v path. Then 8 if i 2 5?1 (k3 ); > > < uu32 if i 2 (3 4 5 )?1 (k2 ) ? 5?1 (k3 ); Qv (i) = > u if i 2 (2 3 4 5 )?1 (k1 ) ? B; > : Qr 1 v (i) otherwise 51
cl-c1
where B = (3 4 5 )?1 (k2 ) [ 5?1 (k3 ). This example shows that for any v, Qv and Qr v are equal everywhere but on the union of nitely many sets of the form (1 : : : n )?1 (k) where s1 ; : : : ; sn ; ck are letters of the root term. In that case (1 : : : n )?1(k) is nite since i[is bounded (i n) and 1 : : : n 2 M . We can conclude that the set ?1 (k) is nite and c1 can be set to be its maximum. ck
2 M ;
is a letter of the root term
Claim 3.69 Let be a witness tree with root r, Q be a valuation of the vertex
cl-c2
set. Then (9c2 2 !)(8 vertex v; v0 )(8i 2 !) v v0 ^ (Qr v )(i) 6= (Qr v0 )(i) ) i c2 :
= max Proof Put c2 def
[
h; 0 i2M =
fi 2 ! j (i) 6= 0 (i)g. This de nition is
transparent since the set, we have to take the maximum of is nite. (M = is nite, fi 2 ! j (i) 6= 0 (i)g is nite if 0 .)
Lemma 3.70 Let be a witness tree with root r, Q be a valuation of the vertex
l-c
set. Then (9c 2 !)(8 vertex v; v0 )(8i 2 !) v v0 ^ Qv (i) 6= Qv0 (i) ) i < c: Proof Put c = maxfc1; c2g +1. Claims 3.68 and 3.69 will prove the statement.
Lemma 3.71 Suppose that Ur is a set with cardinality bigger than 2. Then there is a sequence S 2 ! Ur with the property that for all ; 2 [H ] 6 ) Sr 6 Sr To prove this lemma we will need the following claim. Claim 3.72 There is a set D P (!) for that 2 D ) j j = 2, ; 2 D ^ 6= ) \ = ;, ; 2 [H ]; 6 ) jD \ ff(i); (i)g j i 2 !gj = !. Proof Let hi ; ii (i 2 !) be a series of pairs of transformations of ! with the property that for any ; 2 [H ] with 6 we have that hi ; i i = h; i for in nitely many i 2 !. A series of that kind exists since jH j !; jH j !; j[H ]j jH j !; j[H ] [H ]j !. For all k 2 ! we de ne the set k recursively: 52
l-S
cl-h1
0 = f0(i); 0 (i)g for an i 2 ! with 0 (i) 6= 0 (i). [ k = fk (i); k (i)g for an i 2 ! with k (i) 6= k (i) and i 62 ?k 1 ( j ) [ j
