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DIRECT LIMITS AND REDUCED PRODUCTS OF ALGEBRAS WITH FUZZY EQUALITIES ´ VYCHODIL VILEM

Abstract. We study direct limits and reduced products of algebras with fuzzy equalities. On the one hand, algebras with fuzzy equalities are natural fuzzy structures that disallow to map similar arguments to dissimilar ones. On the other hand, they are exactly the semantic structures of the equational fragment of first-order fuzzy logic. In this paper we propose generalizations of direct limits and reduced products and point out those properties which are not interesting in the classical (bivalent) case, but which seem to be of a crucial importance when considering the quasivarieties of algebras with fuzzy equalities.

1. Introduction There were various efforts to extend the notion of an algebra. For instance, N. Weaver [20, 21] introduced socalled metric algebras which result by equipping a classical algebra with a metric defined on its universe set. The aim of the metric is to express “closeness” of elements. Using the notion of equicontinuity, metric algebras can represent structures where each function maps pairwise close arguments to close results. Our paper is connected with another extension of an algebra which also formalizes the requirement of having functions mapping close elements to close ones but unlike the metric algebras, our extension is developed in the context of fuzzy logic and does not utilize notions like metric and equicontinuity. In the framework of fuzzy logic, one can formalize “closeness” of elements by so-called fuzzy equivalence relation called also similarity relation. The concept of functions preserving similarity leads to the notion of an algebra with fuzzy equality [5]: an algebra with fuzzy equality is a set with operations on it that is equipped with similarity ≈ (a particular fuzzy equivalence relation) such that each operation f is in an appropriate sense compatible with ≈. The compatibility ensures that each f yields similar results if applied to pairwise similar arguments. In addition to the motivation described above, algebras with fuzzy equalities are connected to fuzzy logic in narrow sense (mathematical fuzzy logic) [3, 9, 10, 13, 14, 17]. Namely, algebras with fuzzy equalities represent the semantic (fuzzy) structures of the equational fragment of first-order fuzzy logic. Note that recently fuzzy logic [9, 10, 13, 14] has been profoundly developed. The initial results on algebras with fuzzy equalities [2, 4] showed their nice logical and algebraic properties. Namely, in [2] the author presented a syntactico-semantically complete calculus for reasoning with fuzzy sets of equalities while [4] showed an analogy of the well-known Birkhoff’s variety theorem—varieties of algebras with fuzzy equalities are the model classes of fuzzy sets of identities. The present paper is a continuation of [5], where we introduced the basic structural notions, and it tries to shed more light on the constructions which are vital for the development of quasivariety theory in fuzzy setting, see [6, 19]. In fuzzy logic, an important role is played by the chosen structure of truth degrees (or by a whole class of structures of truth degrees). Most of the results on fuzzy logic use residuated lattices as basic structures of truth degrees (even though there are approaches which use noncommutative monoidal structures, but we will not go into this). Residuated lattices, introduced in the 1930s in ring theory, were introduced into the context of fuzzy logic by Goguen [11, 12]. Fundamental contribution to formal fuzzy logic using residuated lattices as the structures of truth values is due to Pavelka [18]. Later on, various logical calculi were investigated using residuated lattices or particular types of residuated lattices. A thorough information about the role of residuated lattices in fuzzy logic can be obtained from monographs [3, 10, 13, 14]. Recall that a (complete) residuated lattice is an algebra L = hL, ∧, ∨, ⊗, →, 0, 1i of type h2, 2, 2, 2, 0, 0i such that (i) hL, ∧, ∨, 0, 1i is a (complete) lattice with the least element 0 and the greatest element 1, (ii) hL, ⊗, 1i is a commutative monoid, (iii) h⊗, →i is an adjoint pair, i.e. a ⊗ b ≤ c iff a ≤ b → c is valid for each a, b, c ∈ L. 2000 Mathematics Subject Classification. 03B52, 08B05, 08B25. Key words and phrases. direct limit, fuzzy equality, fuzzy logic, reduced product. 1

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Particular types of residuated lattices (distinguishable by identities) include Boolean algebras, Heyting algebras, algebras of Girard’s linear logic, MV-algebras, G¨odel algebras, product algebras, and more generally, BL-algebras (see [14, 15]). In our development we use complete residuated lattices as the basic structures of truth degrees. Let us now stress the main differences between (classical) algebras and algebras with fuzzy equalities. First, when dealing with algebras with fuzzy equalities we use an explicit structure of truth degrees which is generally much weaker than the two-element Boolean algebra being used implicitly in the (classical) universal algebra. Second, an algebra with fuzzy equality has a nontrivial lattice-valued relational part (similarity relation) which satisfies the compatibility condition. Consequently, structural constructions for algebras with fuzzy equalities have to take care of both the functional part and the relational part. Third, the nontrivial relational part and the general structure of truth values allow us to define the notion of a validity degree of an identity. We can thus consider classes of algebras with fuzzy equalities, where certain identities are satisfied to given degrees, etc. Our extension of ordinary algebras can be also useful to get a deeper insight into the classical structural notions—some classical results generalize for algebras with fuzzy equalities in the full scope (i.e. for any complete residuated lattices taken as the structure of truth degrees), however, some results do not. The latter case is especially interesting because one can identify the explicit requirements on structures of truth degrees which are essential. Direct limits and reduced products might be interesting also for the fuzzy logic itself. So far, the research in fuzzy logic has been focused almost exclusively on the aspects motivated by the proof theory (structures of truth degrees, semantic consequence, provability, completeness, etc.) Not much attention has been paid to model-theoretical properties of fuzzy structures. In [8], the authors present a generalization of ultraproducts for structures equipped with relations whose truth degrees form a compact Hausdorff space. In fuzzy case, however, there is only a small effort in studying properties and applications of generalized reduced products. As an exception, in [13] the author presents an approach to ultraproducts in fuzzy setting which is based on the ideas of [8]. An analogous result [23] deals with ultraproducts in the context of Pavelka-style fuzzy logic. In this respect, the present paper contributes to model theory for fuzzy logic. This paper is organized as follows. In Section 2 we present the preliminaries. Section 3 focuses on the direct limits of algebras with fuzzy equalities. In Section 4 we discuss reduced products of algebras with fuzzy equalities and give some remarks on the correlation of the strengthened constructions. 2. Preliminaries In the sequel we briefly recall basic notions of fuzzy logic, fuzzy sets, and algebras with fuzzy equalities. For a detailed description we refer to [5]. In what follows, L always refers to a complete residuated lattice. For L we define notions of an L-set, L-relation, etc. All properties of complete residuated lattices used in the sequel are well known and can be found in any of the above mentioned monographs. Note that the paper contains several examples in which we use the complete residuated L = h[0, 1], min, max, ⊗, →, 0, 1i defined on the real unit interval (unless otherwise stated, the particular definitions of ⊗ and → will not play any role). For brevity we denote L = h[0, 1], min, max, ⊗, →, 0, 1i simply by L = [0, 1]. An L-set A (or fuzzy set with truth degrees in L) on a universe set U is a mapping A : U → L, A(u) ∈ L being interpreted as the truth degree of “element u belongs to A”. For every L-set A : U → L, the support set of A, denoted by Supp(A), is defined by Supp(A) = {u ∈ U | A(u) > 0}. An L-set A is called finite if Supp(A) is finite. For L-sets A and B on the same universe set U we write A ⊆ B iff A(u) ≤ B(u) for each u ∈ U ; and A = B iff A ⊆ B and B ⊆ A. An L-set A in U is called crisp if A(u) ∈ {0, 1} for each u ∈ U . If there is no danger of confusion, we sometimes identify the classical sets with crisp L-sets. A binary L-relation R on U is an L-set on the universe set U × U , i.e. it is a mapping R : U × U → L. A binary L-relation R0 : U × U → L is called a restriction of R : U × U → L, if R0 (u, v) 6= 0 implies R0 (u, v) = R(u, v) for all u, v ∈ U . A restriction R0 of R is called a finite restriction if R0 is a finite L-relation. An L-equivalence (fuzzy equivalence, similarity) relation E on a set U is a binary L-relation on U satisfying E(u, u) = 1 (reflexivity), E(u, v) = E(v, u) (symmetry) and E(u, v) ⊗ E(v, w) ≤ E(u, w) (transitivity) for every u, v, w ∈ U . An L-equivalence on U where E(u, v) = 1 implies u = v is called an L-equality (fuzzy equality). Given an L-equivalence E, the degree E(u, v) ∈ L can be interpreted as the truth degree of proposition “u and v are similar (E-equivalent)”. The above mentioned conditions of reflexivity, symmetry, and transitivity are exactly the semantic representations of the well-known equivalence axioms which are required to be satisfied in degree 1 (the greatest element of L) [3]. In the sequel, we shall define certain algebraic constructions which involve factor sets determined by L-equivalences: for any L-equivalence E on U let 1E be the binary relation on U such that hu, vi ∈ 1E iff E(u, v) = 1. It is easily seen that 1E is a classical equivalence relation. Thus, one may consider the factor set U/1E of U by 1E. For brevity we denote U/1E by U/E and call it the factor set of U by E.

D. LIM. AND R. PROD. OF ALG. WITH FUZZY EQUALITIES

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Moreover, we let [u]E denote the factor class of U by 1E which contains u ∈ U , i.e. [u]E = {v ∈ U | E(u, v) = 1}. For each [u]E and [v]E we put [u]E ≈U/E [v]E = E(u, v). Here ≈U/E is a well-defined L-equality on U/E. Indeed, applying symmetry and transitivity of E, if u0 ∈ [u]E , and v0 ∈ [v]E then E(u, u0 ) = E(v, v0 ) = 1, i.e. E(u, v) = E(u, v) ⊗ E(u, u0 ) ⊗ E(v, v0 ) ≤ E(u0 , v0 ). Analogously, E(u0 , v0 ) ≤ E(u, v). Therefore, ≈U/E is a well-defined binary L-relation on U/E. Moreover, for any u, v ∈ U with [u]E ≈U/E [v]E = 1 we have E(u, v) = 1. That is, u ∈ [v]E , i.e. [u]E = [v]E . The transitivity, symmetry, and reflexivity of ≈U/E follow by properties of E. A mapping f : U n → U , n ∈ N, is compatible with a binary L-relation R on U if for any u1 , v1 , . . . , un , vn ∈ U we have R(u1 , v1 ) ⊗ · · · ⊗ R(un , vn ) ≤ R(f (u1 , . . . , un ), f (v1 , . . . , vn )).

(1)

Compatibility, being the semantic representation of the compatibility (congruence) axiom, has a natural verbal description: it says “if u1 and v1 are R-related and · · · and un and vn are R-related then f (u1 , . . . , un ) and f (v1 , . . . , vn ) are R-related”. We are going to introduce the notion of an algebra with fuzzy equality. As usual, by a type we mean a collection F of function symbols f ∈ F together with their arities. Let t, s, . . . and t ≈ t0 , s ≈ s0 , . . . denote terms (defined as usual) and identities (of a given type F ), respectively. The set of all terms of type F in variables

X will be denoted by T (X).  An algebra with L-equality (shortly an L-algebra) of type F is a triplet M = M, ≈M , F M , where M, F M is an (ordinary) algebra of type F (the so-called skeleton of M) and ≈M is an L-equality on M such that each function f M ∈ F M is compatible with ≈M . For brevity, the skeleton of M will be denoted by ske(M). Before presenting further notions, let us stress the role of compatibility. In the classical case, the compatibility axiom is trivially satisfied. In the fuzzy setting, the compatibility might be thought of as a constraint for operations. Ordinary algebras can be interpreted as L-algebras. For instance, if L is the two-element Boolean algebra, then the notion of an L-algebra coincides with that of an (ordinary) algebra with the usual equality—this way generalizes the results of universal algebra. Another

our approach   way of interpreting an ordinary algebra M, F M as an L-algebra is as follows: we consider M, ≈M , F M such that ≈M is crisp, i.e. {u ≈M v | u, v ∈ M } ⊆ {0, 1} [5]. In classical case, given an algebra M, an identity t ≈ t0 is either valid in M or not. In the fuzzy setting, however, an identity can have a general validity degree (i.e. a truth degree from L), not necessary only 0 or 1. 0 and 1 are but two particular validity degrees, namely, the extremal ones. Given an L-algebra M and a valuation v : X → M , the interpretation ktkM,v of a term t ∈ T (X) in M under v is defined as usual. For an identity t ≈ t0 we define the degree kt ≈ t0 kM,v ∈ L to which t ≈ t0 is true in M under v by kt ≈ t0 kM,v = ktkM,v ≈M kt0 kM,v . 0 Finally, the degree kt ≈ t0 kM ∈ L to which in M is defined using the infimum of truth degrees V t ≈ t is valid 0 ranging over all valuations: kt ≈ t kM = v:X→M kt ≈ t0 kM,v . It is obvious that for L being the two-element Boolean algebra the notion of validity defined in truth degrees coincides with the ordinary one. Example 2.1. (a) If T (X) 6= ∅ then we equip T (X) with functions f T(X) (f ∈ F ) such that f T(X) (t1 , . . . , tn ) = f (t1 , . . . , tn ). Let F T(X) be the collection of all f T(X) ’s. In addition to that, we define a binary L-relation ≈T(X) on T (X) by  1 if t = s, t ≈T(X) s = (2) 0 otherwise.

 Clearly, T(X) = T (X), ≈T(X) , F T(X) is an L-algebra. T(X) is called the term L-algebra of type F in variables X. Even though the relational part of T(X) is crisp, term L-algebras can be used to get L-algebras with nontrivial fuzzy equalities (e.g. by factorization, see below). Term L-algebras play an analogous role as the classical term algebras in universal algebra and will be used in further sections. (b) Let L = h[0, 1], min, max, ⊗, →, 0, 1i be the standard Lukasiewicz algebra [3, 13, 14, 17]. That is, the multiplication ⊗ and residuum → are given by a ⊗ b = max (a + b − 1, 0), and a → b = min (1 − a + b, 1). Consider a type which consists of a single binary function symbol ◦. We define an L-equality ≈M and an operation ◦M on a universe set M = {a, b, c, d, e, f} by the following tables: ≈M a b c d e f

a

b

c

d

e

f

◦M

a

b

c

d

e

f

1

3 4

3 4 3 4

1

3 4 3 4

0 0

0 0

3 4

1

1 4

0

0

1

1 4

1 4

0

0

1 4 1 2 1 4

1 4 1 4 1 2 3 4

a b c d e f

a c c d e f

c b c d e f

c c c d e f

d d d d f f

e e e d e f

f f f f f f

3 4 7 8

1

1 4 7 8 3 4

3 4

1

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´ VILEM VYCHODIL

 One can check that ◦M is compatible with ≈M , i.e. M = M, ≈M , ◦M is an L-algebra. Note that ske(M), being the functional part of M, is idempotent but it is neither associative nor commutative. On the other hand, we have kx ◦ y ≈ y ◦ xkM = kx ◦ (y ◦ z) ≈ (x ◦ y) ◦ zkM = 78 . This can be read: “L-algebra M is commutative and associative in truth degree 78 ”. For more examples we refer the reader to [5]. In [5] we developed algebraic constructions (morphisms, subalgebras, direct products) respecting both the functional and the (non-trivial) relational part of L-algebras. We briefly summarize the constructions involved in the subsequent development. An L-algebra N is a subalgebra of an L-algebra M if (i) ske(N) is a subalgebra of ske(M), and (ii) ≈N is a restriction of ≈M on N . Let M, N be L-algebras. A mapping h : M → N satisfying a ≈M b ≤ h(a) ≈N h(b) (in the order ≤ of L) for all a, b ∈ M is called an ≈-morphism. Thus, an ≈-morphism is a mapping which preserves (is compatible with) the L-equalities (the relational parts of L-algebras). An ≈-morphism h : M → N is called a morphism (of L-algebras), denoted by h : M → N, if h is a morphism of ske(M) and ske(N). Given morphisms h : M → M0 , g : M0 → M00 , the composed mapping (h ◦ g) : M → M00 is a morphism. In much the same way as in the classical case, we distinguish special morphisms of L-algebras. A surjective morphism h : M → N satisfying a ≈M b = h(a) ≈N h(b) for all a, b ∈ M is called an isomorphism. L-algebras M and N are called isomorphic, in symbols M ∼ = N, if there is an isomorphism h : M → N. Observe that for isomorphic L-algebras M and N we have kt ≈ t0 kM = kt ≈ t0 kN for every identity t ≈ t0 . That is, isomorphic L-algebras cannot be distinguished by the (graded) validity of any identity. Trivially, the identical mapping idM : M → M on the universe set of M is an isomorphism. Each mapping h : X → M has a uniquely determined homomorphic extension h] : T(X) → M (the term L-algebra T(X) is defined the same way as in Example 2.1), i.e. h] is a morphism such that h(x) = h] (x) for all x ∈ X. An L-relation θ on M is called a congruence on M if (i) θ is an L-equivalence relation on M , (ii) ≈M ⊆ θ, and (iii) all functions f M ∈ F M are compatible with θ. The congruences on M form a complete lattice the least and the greatest elements of which are ≈M , and θ such that θ(a, b) = 1 (a, b ∈ M ), respectively. For brevity, we denote the greatest congruence on M simply by M × M . For a binary L-relation R on

M we denote byθ(R) the congruence generated by R. Given a congruence θ on M, the L-algebra M/θ = M/θ, ≈M/θ , F M/θ is called the factor L-algebra of M modulo θ, if (i) ske(M/θ) is the factor algebra of ske(M) modulo 1 θ, and (ii) [a]θ ≈M/θ [b]θ = θ(a, b) for all a, b ∈ M . For a morphism h : M → N we define a congruence θh on M by θh (a, b) = h(a) ≈N h(b). For a congruence θ on M, a morphism hθ : M → M/θ defined by hθ (a) = [a]θ (a ∈ M ) is called morphism.Q Given a system {Mi | i ∈ I} of

Qa natural Q Q Q i∈I Mi , where (i) ske( i∈I Mi , F L-algebras, a direct product is an L-algebra i∈I M = M , ≈ i i i∈I Mi ) i∈I Q Q V Q is the direct product i∈I ske(Mi ), and (ii) a ≈ i∈I Mi b = i∈I a(i) ≈Mi b(i) for all a, b ∈ i∈I M i . The properties of morphisms and congruences of L-algebras are analogous to those of their classical counterparts. For instance, the well-known isomorphism theorems hold for L-algebras in the full scope [3, 5]. On the other hand, some properties of algebras generalize only for particular subclasses of complete residuated lattices. This is the case of e.g. subdirect representation [5]. Nevertheless, in the sequel we shall use another representation of L-algebras. Let T(X) be a term L-algebra (see Example 2.1), R be a binary L-relation on T (X). Each L-algebra M such that M ∼ = T(X)/θ(R) is said to be presented by hX, Ri. Moreover, M is called finitely presented if X and R can be chosen so that X is a finite set and R is a finite L-relation. Note that each L-algebra M is presented by hX, Ri, where X is a suitably large set of variables and R is a binary L-relation on T (X). Indeed, one can consider |X| ≥ |M |, and a surjective mapping h : X → M . Then for the homomorphic extension h] : T(X) → M of h we have M ∼ = T(X)/θh] due to the first isomorphism theorem [3, 5]. That is, M is presented by hX, θh] i. Remark 2.2. Let us comment some more on the role of complete residuated lattice as the structures of truth degrees. In the definition of algebras with fuzzy equalities, we used truth degrees from a lattice ordered structure (of truth degrees) to express similarity of elements. The condition of compatibility with functions (1) was defined using the multiplication ⊗ which can be seen as a particular interpretation of logical connective “conjunction”. This suggests that our structures of truth degrees should be particular lattice-ordered monoids. Moreover, in order to consider direct products of algebras with fuzzy equalities, we assumedQthat the lattice Q order is complete (otherwise a ≈ i∈I Mi b would not be defined in general, see the definition of i∈I Mi ). Thus, our structures of truth degrees should be complete lattice-ordered monoinds. Each complete residuated lattice L = hL, ∧, ∨, ⊗, →, 0, 1i fulfills this requirement. On the other hand, one may ask if we actually need the binary operation of residuum → (interpretation of logical connective “implication”). Even if → has not been (explicitly) used so far, it is important. For instance, in order to have desirable properties of several algebraic constructions,

D. LIM. AND R. PROD. OF ALG. WITH FUZZY EQUALITIES

W we need the following relationship between ⊗ and : W W a ⊗ i∈I bi = i∈I (a ⊗ bi ).

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(3)

Now, the following assertion is true (see [3]): if hL, ∧, ∨, 0, 1i is a complete lattice, and hL, ⊗, 1i is a commutative monoid (i.e., ⊗ is associative, commutative, and a ⊗ 1 = 1 ⊗ a = a, where 1 the greatest element of L with respect to the lattice order ≤), then there exists a residuum → suchWthat h⊗, →i is an adjoint pair iff (3) is true for any a ∈ L and {bi ∈ L | i ∈ I} (namely, one can put a → b = {c ∈ L | a ⊗ c ≤ b}). Since the required property (3) is equivalent to the existence of residuum, we develop algebras with fuzzy equalities over complete residuated lattices. In further sections we will take advantage of several properties of residuated lattices which can be found in [3, 10, 13, 14]. 3. Direct limits We begin our development with the definition of a weak direct family which generalizes the notion of a direct family known from the (classical) universal algebra. In addition to that, we introduce its strengthened form by postulating an additional condition (which holds true automatically in the ordinary case). Having generalized the basic notions, we introduce direct limits and analyze their properties. Definition 3.1. A weak direct family (of L-algebras) consists of: (i) a directed index set hI, ≤i, i.e. I 6= ∅, and for every i, j ∈ I there is k ∈ I such that i, j ≤ k; (ii) a family {Mi | i ∈ I} of pairwise disjoint L-algebras; (iii) a family {hij : Mi → Mj | i ≤ j} of morphisms, where hii = idMi

for every i ∈ I,

(4)

hik = hij ◦ hjk

for all i, j, k ∈ I, where i ≤ j ≤ k.

(5)

A weak direct family is called a direct family if for every a ∈ Mi , and b ∈ Mj there exists k ∈ I such that i, j ≤ k, and for each l ∈ I with k ≤ l we have hik (a) ≈Mk hjk (b) = hil (a) ≈Ml hjl (b).

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Remark 3.2. (a) A (weak) direct family of L-algebras will be denoted simply by {Mi | i ∈ I}. If there is no danger of confusion, we will not mention the morphisms hij : Mi → Mj explicitly. (b) In general, there are weak direct families which do not satisfy (6).

Take L = [0,1] as the structure of truth degrees and a family {Mi | i ∈ [0, 1)} of L-algebras, where Mi = {ai , bi }, ≈Mi , ∅ , and ai ≈Mi bi = i. Moreover, morphisms hij : Mi → Mj (i ≤ j) defined by hij (ai ) = aj , hij (bi ) = bj evidently satisfy (4) and (5). Therefore, S h[0, 1), ≤i together with {Mi | i ∈ [0, 1)} and hij ’s is a weak direct family. On the other hand, for ai , bj ∈ m∈[0,1) Mm , and every k ≥ i, j there is l > k such that hik (ai ) ≈Mk hjk (bj ) = ak ≈Mk bk = k < l = al ≈Ml bl = hil (ai ) ≈Ml hjl (bj ), showing that {Mi | i ∈ [0, 1)} is not a direct family. Lemma 3.3. Let {Mi | i ∈ I} be a weak direct family. For every a ∈ Mi , b ∈ Mj , and arbitrary indices k, l ∈ I such that i, j ≤ k ≤ l we have hik (a) ≈Mk hjk (b) ≤ hil (a) ≈Ml hjl (b), W Mk hjk (b) = m≥l him (a) ≈Mm hjm (b). k≥i,j hik (a) ≈

W

(7) (8)

Proof. (7): Using (5) we have hik (a) ≈Mk hjk (b) ≤ hkl (hik (a)) ≈Ml hkl (hjk (b)) = hil (a) ≈Ml hjl (b). (8): Take an index k0 ≥ i, j. For m0 ∈ I such that m0 ≥ k0 , l, we can use (7) to get W hik0 (a) ≈Mk0 hjk0 (b) ≤ him0 (a) ≈Mm0 hjm0 (b) ≤ m≥l him (a) ≈Mm hjm (b), by which follows the “≤”-part of (8). The converse inequality holds trivially.



Remark 3.4. Let L be a complete residuated lattice, where each a ∈ L is compact (i.e., L is a noetherian lattice, see [7]). Then every weak direct family is a direct family. W Indeed, the compactness yields Wn that for any a ∈ Mi , and b ∈ Mj , there are indices k1 , . . . , kn ≥ i, j such that k≥i,j hik (a) ≈Mk hjk (b) = m=1 hikm (a) ≈Mkm hjkm (b). Thus, take an index k ∈ I with k ≥ k1 , . . . , kn . Now (7) gives hikm (a) ≈Mkm hjkm (b) ≤ hik (a) ≈Mk hjk (b) for each m = 1, . . . , n. Therefore, hik (a) ≈Mk hjk (b) is the greatest one of all hik0 (a) ≈Mk0 hjk0 (b) for k 0 ≥ i, j. Since for each l ≥ k we have hik (a) ≈Mk hjk (b) ≤ hil (a) ≈Ml hjl (b) by (7), it follows that in fact hik (a) ≈Mk hjk (b) = hil (a) ≈Ml hjl (b) for all l ≥ k proving (6).

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Definition 3.5. For a weak direct family {Mi | i ∈ I} let θ∞ denote the binary L-relation on by W θ∞ (a, b) = k≥i,j hik (a) ≈Mk hjk (b)

S

i∈I

Mi defined (9)

for all a ∈ Mi , b ∈ Mj . Remark 3.6. If {Mi | i ∈ I} is a direct family, then (9) can be expressed equivalently without using the general suprema. Indeed, taking into account (6) and (8), for all a ∈ Mi , b ∈ Mj there is k0 ≥ i, j such that W W θ∞ (a, b) = k≥i,j hik (a) ≈Mk hjk (b) = m≥k0 him (a) ≈Mm hjm (b) = hik0 (a) ≈Mk0 hjk0 (b). Lemma 3.7. Let {Mi | i ∈ I} be a weak direct family. The following are properties of θ∞ :
(i) θ∞ (a, b) = θ∞ hil (a), hjl (b)S for all a ∈ Mi , b ∈ Mj , and l ≥ i, j; (ii) θ∞ is an L-equivalence on i∈I Mi ;
(iii) θ∞ a, hik (a) = 1 for every a ∈ Mi , and k ≥ i; (iv) for every n-ary f ∈ F , a1 ∈ Mi1 , b1 ∈ Mj1 , . . . , an ∈ Min , bn ∈ Mjn , and k ≥ i1 , j1 , . . . , in , jn we have
θ∞ (a1 , b1 ) ⊗ · · · ⊗ θ∞ (an , bn ) ≤ θ∞ f Mk (hi1 k (a1 ), . . . , hin k (an )), f Mk (hj1 k (b1 ), . . . , hjn k (bn )) . Proof. (i): Clearly, (5) and (8) give W W θ∞ (a, b) = k≥i,j hik (a) ≈Mk hjk (b) = m≥l him (a) ≈Mm hjm (b) = W = m≥l hlm (hil (a)) ≈Mm hlm (hjl (b)) = θ∞ (hil (a), hjl (b)). W (ii): We have θ∞ (a, a) = k≥i hik (a) ≈Mk hik (a) = 1 for every a ∈ Mi , i.e. θ∞ is reflexive. Symmetry is obvious. It remains to check transitivity. Let a ∈ Mi , b ∈ Mj , c ∈ Mk , and let l ≥ i, j, k. Furthermore, (7) and (i) together with the monotony of ⊗ yield

θ∞ (a, b) ⊗ θ∞ (b, c) = θ∞ (hil a), hjl (b) ⊗ θ∞ (hjl b), hkl (c) = W W = m≥l him (a) ≈Mm hjm (b) ⊗ n≥l hjn (b) ≈Mn hkn (c) =
W = m,n≥l him (a) ≈Mm hjm (b) ⊗ hjn (b) ≈Mn hkn (c) =
W W = n≥l hin (a) ≈Mn hjn (b) ⊗ hjn (b) ≈Mn hkn (c) ≤ n≥l hin (a) ≈Mn hkn (c) = θ∞ (a, c). Hence, θ∞ is an L-equivalence. (iii): Let us have a ∈ Mi , k ≥ i. Take l ∈ I such that l ≥ k, i. The reflexivity of θ∞ together with (i) give
θ∞ (a, hik (a)) = θ∞ hil (a), hkl (hik (a)) = θ∞ (hil (a), hil (a)) = 1. (iv): For an n-ary f ∈ F , arbitrary am ∈ Mim , bm ∈ Mjm (m = 1, . . . , n), and k ≥ i1 , j1 , . . . , in , jn , we can use the compatibility of f Ml with ≈Ml to get Nn W θ∞ (a1 , b1 ) ⊗ · · · ⊗ θ∞ (an , bn ) = m=1 km ≥k him km (am ) ≈Mkm hjm km (bm ) = W Nn W Nn = k1 ,...,km ≥k m=1 him km (am ) ≈Mkm hjm km (bm ) = l≥k m=1 him l (am ) ≈Ml hjm l (bm ) ≤

W ≤ l≥k f Ml hi1 l (a1 ), . . . , hin l (an ) ≈Ml f Ml hj1 l (b1 ), . . . , hjn l (bn ) =

W = l≥k f Ml hkl (hi1 k (a1 )), . . . , hkl (hin k (an )) ≈Ml f Ml hkl (hj1 k (b1 )), . . . , hkl (hjn k (bn )) =

W = l≥k hkl f Mk (hi1 k (a1 ), . . . , hin k (an )) ≈Ml hkl f Mk (hj1 k (b1 ), . . . , hjn k (bn )) =
= θ∞ f Mk (hi1 k (a1 ), . . . , hin k (an )), f Mk (hj1 k (b1 ), . . . , hjn k (bn )) , which is the desired inequality.



Condition (iv) of Lemma 3.7 is similar to that of compatibility, but in this case, (iv) expressesS a compatibility with respect to homomorphic images. Now we define suitable operations on the factorization of i∈I Mi by θ∞ .

S
 lim Mi Definition 3.8. Let {Mi | i ∈ I} be a (weak) direct family. lim Mi = , F lim Mi , where i∈I Mi /θ∞ , ≈
S S (i) i∈I Mi /θ∞ is a factorization of i∈I Mi by θ∞ , S
(ii) [a]θ∞ ≈lim Mi [b]θ∞ = θ∞ (a, b) for all [a]θ∞ , [b]θ∞ ∈ Mi /θ∞ i∈I
  (iii) f lim Mi [a1 ]θ∞ , . . . , [an ]θ∞ = f Mk (hi1 k (a1 ), . . . , hin k (an )) θ for every n-ary f ∈ F , arbitrary ∞
S [a1 ]θ∞ , . . . , [an ]θ∞ ∈ i∈I Mi /θ∞ with a1 ∈ Mi1 , . . . , an ∈ Min , and k ∈ I such that k ≥ i1 , . . . , in , is called a direct limit of a (weak ) direct family {Mi | i ∈ I}.

D. LIM. AND R. PROD. OF ALG. WITH FUZZY EQUALITIES

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Remark 3.9. A direct limit lim Mi of a weak direct family {Mi | i ∈ I} is an L-algebra. Obviously, ≈lim Mi is an L-equality. It remains to showthat each f lim Mi is well defined and compatible with ≈lim Mi . First, we show  M k that f (hi1 k (a1 ), . . . , hin k (an )) θ∞ given by (iii) does not depend on the chosen k ∈ I. Thus, take k 0 ∈ I with k 0 ≥ i1 , . . . , in , and arbitrary l ≥ k, k 0 . Lemma 3.7 gives
θ∞ f Mk (hi1 k (a1 ), . . . , hin k (an )), f Ml (hi1 l (a1 ), . . . , hin l (an )) =

= θ∞ f Mk (hi1 k (a1 ), . . . , hin k (an )), hkl f Mk (hi1 k (a1 ), . . . , hin k (an )) = 1.     That is, f Mk (hi1 k (a1 ), . . . , hin k (an )) θ∞ = f Ml (hi1 l (a1 ), . . . , hin l (an )) θ∞ and analogously for k 0 . Hence,   M   f k (hi1 k (a1 ), . . . , hin k (an )) θ∞ = f Mk0 (hi1 k0 (a1 ), . . . , hin k0 (an )) θ∞ .
Moreover, f lim Mi [a1 ]θ∞ , . . . , [an ]θ∞ does not depend on a1 , . . . , an chosen from classes [a1 ]θ∞ , . . . , [an ]θ∞ , because for bm ∈ [am ]θ∞ , bm ∈ Mjm (m = 1, . . . , n), and k ≥ i1 , j1 , . . . , in , jn we have
1 = θ∞ (a1 , b1 ) ⊗ · · · ⊗ θ∞ (an , bn ) ≤ θ∞ f Mk (hi1 k (a1 ), . . . , hin k (an )), f Mk (hj1 k (b1 ), . . . , hjn k (bn )) =     = f Mk (hi1 k (a1 ), . . . , hin k (an )) θ ≈lim Mi f Mk (hj1 k (b1 ), . . . , hjn k (bn )) θ . ∞ ∞     Therefore, f Mk (hi1 k (a1 ), . . . , hin k (an )) θ = f Mk (hj1 k (b1 ), . . . , hjn k (bn )) θ , i.e. f lim Mi is well defined. ∞ ∞ It remains to check the compatibility. Take f lim Mi ∈ F lim Mi and [a1 ]θ∞ , [b1 ]θ∞ , . . . , [an ]θ∞ , [bn ]θ∞ ∈
S i∈I Mi /θ∞ , where am ∈ Mim , bm ∈ Mjm (m = 1, . . . , n). For k ∈ I such that k ≥ i1 , j1 , . . . , in , jn , Lemma 3.7 together with the definition of ≈lim Mi yield [a1 ]θ∞ ≈lim Mi [b1 ]θ∞ ⊗ · · · ⊗ [an ]θ∞ ≈lim Mi [bn ]θ∞ = θ∞ (a1 , b1 ) ⊗ · · · ⊗ θ∞ (an , bn ) ≤
≤ θ∞ f Mk (hi1 k (a1 ), . . . , hin k (an )), f Mk (hj1 k (b1 ), . . . , hjn k (bn )) =     = f Mk (hi1 k (a1 ), . . . , hin k (an )) θ∞ ≈lim Mi f Mk (hj1 k (b1 ), . . . , hjn k (bn )) θ∞ =

= f lim Mi [a1 ]θ∞ , . . . , [an ]θ∞ ≈lim Mi f lim Mi [b1 ]θ∞ , . . . , [bn ]θ∞ . Hence, lim Mi is a well-defined L-algebra. lim Mi Lemma 3.10. Let lim , and S Mi be
the weak direct limit of a family {Mi | i ∈ I}. Then for every n-ary f M /θ we have [a1 ]θ∞ , . . . , [an ]θ∞ ∈ i ∞ i∈I
  lim Mi f [a1 ]θ∞ , . . . , [an ]θ∞ = f Mk (a01 , . . . , a0n ) θ∞ ,

where a01 , . . . , a0n ∈ Mk , and a0m ∈ [am ]θ∞ (m = 1, . . . , n). Proof. Since θ∞ (am , a0m ) = 1 for each m = 1, . . . , n, we can take an index l ∈ I such that l ≥ k, i1 , . . . , in and apply Definition 3.8 and Lemma 3.7:  M 0  
   f k (a1 , . . . , a0n ) θ∞ = hkl f Mk (a01 , . . . , a0n ) θ∞ = f Ml (hkl (a01 ), . . . , hkl (a0n )) θ∞ = 
 = f Ml (hi1 l (a1 ), . . . , hin l (an )) θ∞ = f lim Mi [a1 ]θ∞ , . . . , [an ]θ∞ , proving the assertion.



Definition 3.11. Let {Mi | i ∈ I} be a weak direct family. A family {hi : Mi → lim Mi | i ∈ I} of morphisms, where hi (a) = [a]θ∞ (i ∈ I, a ∈ Mi ) is called the limit cone of the weak direct family {Mi | i ∈ I}. Let N be an L-algebra. A family {gi : Mi → N | i ∈ I} of morphisms is said to satisfy the direct limit property (DLP) with respect to {Mi | i ∈ I} if gi = hij ◦ gj for all i ≤ j and for every family {gi0 : Mi → N0 | i ∈ I} of morphisms with gi0 = hij ◦ gj0 for all i ≤ j, there exists a unique morphism g : N → N0 such that gi0 = gi ◦ g for every i ∈ I. Remark 3.12. (a) Every hi : Mi → lim Mi of the limit cone of {Mi | i ∈ I} is indeed a morphism. Clearly, for all a, b ∈ Mi (i ∈ I) we have W a ≈Mi b ≤ k≥i hik (a) ≈Mk hik (b) = θ∞ (a, b) = [a]θ∞ ≈lim Mi [b]θ∞ = hi (a) ≈lim Mi hi (b). Furthermore, for an n-ary f ∈ F , and a1 , . . . , an ∈ Mi :
    hi f Mi (a1 , . . . , an ) = f Mi (a1 , . . . , an ) θ∞ = f Mi (hii (a1 ), . . . , hii (an )) θ∞ =

= f lim Mi [a1 ]θ∞ , . . . , [an ]θ∞ = f lim Mi hi (a1 ), . . . , hi (an ) , i.e. hi is a morphism. (iii) of Lemma 3.7 gives hi (a) = [a]θ∞ = [hij (a)]θ∞ = hj (hij (a)), i.e. hi = hij ◦ hj .

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(b) If hI, ≤i is a finite directed index set, then I has the greatest element. Therefore, every weak direct family {Mi | i ∈ I} is a direct family since for every a ∈ Mi , and b ∈ Mj , (6) is satisfied trivially for k being the greatest element of I. Consequently, θ∞ (a, b) = hik (a) ≈Mk hjk (b). Moreover, for hk : Mk → lim Mi we have a ≈Mk b = hkk (a) ≈Mk hkk (b) = θ∞ (a, b) = hk (a) ≈lim Mi hk (b),
S and for every [c]θ∞ ∈ i∈I Mi /θ∞ with c ∈ Mi we have hk (hik (c)) = [hik (c)]θ∞ = [c]θ∞ . Hence, hk is an isomorphism (hk is compatible with operations since it is a part of the limit cone), Mk ∼ = lim Mi . In other words, the direct limit is trivial for finite hI, ≤i. Theorem 3.13. Let {Mi | i ∈ I} be a weak direct family. Then the limit cone of {Mi | i ∈ I} satisfies DLP with respect to {Mi | i ∈ I}. Proof. Let {hi : Mi → lim Mi | i ∈ I} be the limit cone of {Mi | i ∈ I}. Take a family {gi : Mi → N | i ∈ I} of morphisms such that gi = hij ◦gj (i ≤ j). We check the existence and uniqueness of a morphism h : lim Mi → N, where gi = hi ◦ h (i
S S ∈ I). First, each a ∈ i∈I Mi belongs to some Mi . Hence, define h : i∈I Mi /θ∞ → N by
h [a]θ∞ ) = gi (a , where a ∈ Mi . (10) For every a ∈ Mi and b ∈ Mj we have W [a]θ∞ ≈lim Mi [b]θ∞ = θ∞ (a, b) = k≥i,j hik (a) ≈Mk hjk (b) ≤ W W ≤ k≥i,j gk (hik (a)) ≈N gk (hjk (b)) = k≥i,j gi (a) ≈N gj (b) = gi (a) ≈N gj (b). Thus, [a]θ∞ = [b]θ∞ implies gi (a) = gj (b). As a consequence, h defined by (10) is a well-defined ≈-morphism.
S Now for any n-ary f ∈ F , and [a1 ]θ∞ , . . . , [an ]θ∞ ∈ i∈I Mi /θ∞ there are indices i1 , . . . , in , l ∈ I such that i1 , . . . , in ≤ l, and a1 ∈ Mi1 , . . . , an ∈ Min . Using Lemma 3.7, it follows that


h f lim Mi ([a1 ]θ∞ , . . . , [an ]θ∞ ) = h f Ml hi1 l (a1 ), . . . , hin l (an ) θ∞ =

= gl f Ml (hi1 l (a1 ), . . . , hin l (an )) = f N gl (hi1 l (a1 )), . . . , gl (hin l (an )) =

= f N gi1 (a1 ), . . . , gin (an ) = f N h([a1 ]θ∞ ), . . . , h([an ]θ∞ ) . That is, h : lim Mi → N is the required morphism with gi (a) = h([a]θ∞ ) = h(hi (a)) = (hi ◦ h)(a). S Finally, let h0 : lim M
i → N be a morphism satisfying gi = hi ◦ h
0 . Every a ∈ i∈I Mi belongs to Mi for some  i ∈ I. That is, h0 [a]θ∞ = h0 (hi (a)) = gi (a) = h(hi (a)) = h [a]θ∞ , proving the uniqueness. Theorem 3.14. Let {Mi | i ∈ I} be a weak direct family and let N be an L-algebra. Then there is a family {gi : Mi → N | i ∈ I} of morphisms satisfying DLP w.r.t. {Mi | i ∈ I} iff N ∼ = lim Mi . Proof. The proof is analogous to that one known from the ordinary case, so we give only a sketch. “⇒”: For N being lim Mi , the limit cone {hi : Mi → lim Mi | i ∈ I} satisfies DLP w.r.t. {Mi | i ∈ I}. Hence, it suffices to prove that for any families of morphisms {gi : Mi → N | i ∈ I}, {gi0 : Mi → N0 | i ∈ I}, satisfying DLP w.r.t. {Mi | i ∈ I} we have N ∼ = N0 . Since both families satisfy DLP w.r.t. {Mi | i ∈ I}, there are uniquely determined morphisms g : N → N0 , g 0 : N0 → N, where gi = gi0 ◦ g 0 , gi0 = gi ◦ g. Consequently gi = gi0 ◦ g 0 = (gi ◦ g) ◦ g 0 = gi ◦ (g ◦ g 0 ). Thus, g ◦ g 0 = idN . Analogously, g 0 ◦ g = idN 0 . Using the morphism theorems [5], one can conclude that g, g 0 are mutually inverse isomorphisms between N and N0 . That is, for N0 being lim Mi we have N ∼ = lim Mi . “⇐”: For N ∼ = lim Mi we can take morphisms gi : Mi → N such that gi = hi ◦ h with h : lim Mi → N being an isomorphism. It is routine to check that {gi : Mi → N} satisfies DLP w.r.t. {Mi | i ∈ I}.  In the ordinary case, any algebra is isomorphic to a direct limit of finitely presented algebras. The following theorem presents an analogous characterization for L-algebras. Theorem 3.15. Every L-algebra is isomorphic to a direct limit of a direct family of finitely presented L-algebras. Proof. Let M be an L-algebra. For brevity, we identify M with T(X)/θ(R), where X is a set of variables and R is a binary L-relation of T (X). Recall that for every Y ⊆ X we can consider a restriction θ(R)|T(Y ) of θ(R) on T(Y ). Now let us have the index set  I = hY, Si | Y ⊆ X, Y is finite, S is a finite restriction of θ(R)|T(Y ) .

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We can define a partial order ≤ on I by hYi , Si i ≤ hYj , Sj i

iff Yi ⊆ Yj and Si ⊆ Sj .

Obviously, hI, ≤i is directed. For brevity, we denote indices of the form hYi , Si i, hYj , Sj i , . .
. simply by i, j, . . . We introduce morphisms hij : T(Yi )/θ(Si ) → T(Yj )/θ(Sj ) (i ≤ j) defined by hij [t]θ(Si ) = [t]θ(Sj ) . Clearly, every hij satisfies (4) and (5). Thus, hI, ≤i, {T(Yi )/θ(Si ) | i ∈ I} together with hij ’s is a weak direct family. Moreover, it is even a direct family. Indeed, take [ti ]θ(Si ) ∈ T(Yi )/θ(Si ), [tj ]θ(Sj ) ∈ T(Yj )/θ(Sj ). There is k ≥ i, j such that Yk = Yi ∪ Yj and Sk (ti , tj ) = θ(R)(ti , tj ). Clearly, for every l ≥ k we have

hik [ti ]θ(Si ) ≈T(Yk )/θ(Sk ) hjk [tj ]θ(Sj ) = [ti ]θ(Sk ) ≈T(Yk )/θ(Sk ) [tj ]θ(Sk ) = θ(Sk )(ti , tj ) = θ(R)(ti , tj ) =

= θ(Sl )(ti , tj ) = [ti ]θ(Sl ) ≈T(Yl )/θ(Sl ) [tj ]θ(Sl ) = hil [ti ]θ(Si ) ≈T(Yl )/θ(Sl ) hjl [tj ]θ(Sj ) , showing that hI, ≤i, {T(Yi )/θ(Si ) | i ∈ I} together with hij ’s is a direct family. The proof is finished by showing that there is a family {hi : T(Yi )/θ(Si ) → T(X)/θ(R) | i ∈ I} of morphisms satisfying DLP w.r.t. {T(Yi )/θ(Si ) | i ∈ I}. Then T(X)/θ(R) ∼ = lim T(Yi )/θ(Si ) on account of Theorem 3.14. In the rest of the
proof, we denote by var(t) the set of all variables occurring in the term t. Put hi [t]θ(Si ) = [t]θ(R) for every t ∈ T (Yi ). Each hi is a morphism, and hi = hij ◦ hj (i ≤ j). Take a family {gi : T(Yi )/θ(Si ) → N | i ∈ I} of morphisms with gi = hij ◦ gj (i ≤ j), and let us define h : T(X)/θ(R) → N by

h [t]θ(R) = gi [t]θ(Si ) ,
where t ∈ T (X), and i ∈ I such that var(t) ⊆ Yi . Note that h [t]θ(R) does not depend on the choice of i ∈ I since for i, j ∈ I with var(t) ⊆ Yi , Yj we can take l ≥ i, j, and then







gi [t]θ(Si ) ≈N gj [t]θ(Sj ) = gl hil [t]θ(Si ) ≈N gl hjl [t]θ(Sj ) = gl [t]θ(Sl ) ≈N gl [t]θ(Sl ) = 1

yields gi [t]θ(Si ) = gj [t]θ(Sj ) . Now take ti , tj ∈ T (X) with var(ti ) ⊆ Yi , and var(tj ) ⊆ Yj . For k ∈ I, k ≥ i, j such that θ(R)(ti , tj ) = θ(Sk )(ti , tj ), it follows that [ti ]θ(R) ≈T(X)/θ(R) [tj ]θ(R) = θ(R)(ti , tj ) = θ(Sk )(ti , tj ) = [ti ]θ(Sk ) ≈T(Yk )/θ(Sk ) [tj ]θ(Sk ) ≤







≤ gk [ti ]θ(Sk ) ≈N gk [tj ]θ(Sk ) = gk hik [ti ]θ(Si ) ≈N gk hjk [tj ]θ(Sj ) = gi [ti ]θ(Si ) ≈N gj [tj ]θ(Sj ) .

Thus, [ti ]θ(R) = [tj ]θ(R) implies gi [ti ]θ(Si ) = gj [tj ]θ(Sj ) . Altogehter, h is a well-defined ≈-morphism. For any n-ary f ∈ F , and t1 , . . . , tn ∈ T (X), there is k ∈ I such that var(ti ) ⊆ Yk (i = 1, . . . , n). We have

h f T(X)/θ(R) ([t1 ]θ(R) , . . . , [tn ]θ(R) ) = h [f (t1 , . . . , tn )]θ(R) =

= gk [f (t1 , . . . , tn )]θ(Sk ) = gk f T(Yk )/θ(Sk ) ([t1 ]θ(Sk ) , . . . , [tn ]θ(Sk ) ) =





= f N gk [t1 ]θ(Sk ) , . . . , gk [t1 ]θ(Sk ) = f N h [t1 ]θ(R) , . . . , h [tn ]θ(R) .



Hence, h is a morphism. In addition to that, gi [t]θ(Si ) = h [t]θ(R) = h hi [t]θ(Si ) , for every t ∈ T (Yi ), i.e. gi = hi ◦ h. Finally, we check the uniqueness of h. Let h0 : T(X)/θ(R) → N be satisfying gi = hi
◦ h0


a morphism 0 0 (i ∈ I). It is immediate that for t ∈ T (X), var(t) ⊆ Yi we have h [t]θ(R) = h hi [t]θ(Si ) = gi [t]θ(Si ) =


h hi [t]θ(Si ) = h [t]θ(R) . Altogether, h : T(X)/θ(R) → N is a unique morphism with gi = hi ◦ h (i ≤ j).  Remark 3.16. Using the generalization of direct unions of L-algebras [5], one can show (by standard arguments) that every weak direct limit is isomorphic to a direct union of a directed family of L-algebras. Theorem 3.17. Let {Mi | ∈ I} be a direct family and let {hi : Mi → lim Mi | i ∈ I} be the limit cone of lim Mi . Suppose h : N → lim Mi is a morphism, where N is a finitely presented L-algebra. Then there exists k ∈ I and a morphism g : N → Mk such that h = g ◦ hk . Proof. Since N is supposed to be finitely presented, we can identify N with some T(X)/θ(R), where X and R are finite. Thus, let us assume a morphism h : T(X)/θ(R) → lim Mi is given. It is obvious that T(X)/θ(R)
is generated [5] by [x]θ(R) | x ∈ X . For every variable x ∈ X there is an index ix ∈ I such that h [x]θ(R) ∈ hix (Mix ). Since there are only finitely many variables in X, we can choose j ∈ I with j ≥ ix (x ∈ X). Clearly,
h [x]θ(R) ∈ hix (Mix ) = hj (hix j (Mix )) ⊆ hj (Mj ) for each x ∈ X. Therefore, h(hθ(R) (X)) ⊆ hj (Mj ), where hθ(R) : T(X) → T(X)/θ(R) is a natural morphism.
Following this observation, for each x ∈ X there is ax ∈ Mj such that h [x]θ(R) = hj (ax ) ∈ hj (Mj ). Hence, we

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introduce a mapping v : X → Mj by putting v(x) = ax (x ∈ X). By definition, hj (v(x)) = h [x]θ(R) for each
x ∈ X. Since for v ] we have hj (v ] (t)) = h [t]θ(R) for all t ∈ T (X), it follows that v ] ◦ hj = hθ(R) ◦ h. Recall that R is finite, i.e. Supp(R) = {ht1 , t01 i , . . . , htm , t0m i}. Since {Mi | ∈ I} is a direct family, (6) yields that for each i = 1, . . . , m there is ki ∈ I such that θ∞ (v ] (ti ), v ] (t0i )) = hjki (v ] (ti )) ≈Mki hjki (v ] (t0i )), see Remark 3.6. Following this observation, for k ≥ k1 , . . . , km we have

θ(R)(ti , t0i ) = [ti ]θ(R) ≈T(X)/θ(R) [t0i ]θ(R) ≤ h [ti ]θ(R) ≈lim Mi h [t0i ]θ(R) = hj (v ] (ti )) ≈lim Mi hj (v ] (t0i )) = = θ∞ (v ] (ti ), v ] (t0i )) = hjki (v ] (ti )) ≈Mki hjki (v ] (t0i )) ≤ hjk (v ] (ti )) ≈Mk hjk (v ] (t0i )). Thus, R(ti , t0i ) ≤ hjk (v ] (ti )) ≈Mk hjk (v ] (t0i )) = θv] ◦hjk (ti , t0i ) for each i = 1, . . . , m. Since θv] ◦hjk is a congruence
and θ(R) is generated by R, we get θ(R) ⊆ θv] ◦hjk . Finally, put g [t]θ(R) = hjk (v ] (t)). For t, t0 ∈ T (X) we have

[t]θ(R) ≈T(X)/θ(R) [t0 ]θ(R) = θ(R)(t, t0 ) ≤ θv] ◦hjk (t, t0 ) = hjk (v ] (t)) ≈Mk hjk (v ] (t0 )) = g [t]θ(R) ≈Mk g [t0 ]θ(R) , i.e. g is a well-defined ≈-morphism. For any n-ary f ∈ F , and [t1 ]θ(R) , . . . , [tn ]θ(R) ∈ T (X)/θ(R) we have




g f T(X)/θ(R) [t1 ]θ(R) , . . . , [tn ]θ(R) = g [f (t1 , . . . , tn )]θ(R) = hjk v ] f (t1 , . . . , tn ) =



= hjk f Mj v ] (t1 ), . . . , v ] (tn ) = f Mk hjk (v ] (t1 )), . . . , hjk (v ] (tn )) = f Mk g([t1 ]θ(R) ), . . . , g([tn ]θ(R) ) . Hence, g : T(X)/θ(R) → Mk is a morphism. In addition to that,



h [t]θ(R) = hj (v ] (t)) = hk (hjk (v ] (t))) = hk g [t]θ(R) = (g ◦ hk ) [t]θ(R) holds for all [t]θ(R) ∈ T (X)/θ(R), i.e. h = g ◦ hk .



Remark 3.18. The existence of the morphism given by Theorem 3.17 is limited to direct families. In the bivalent case, every weak direct family is a direct family, thus Theorem 3.17 coincides with the well-known image factorization theorem for the ordinary algebras. The following example illustrates that postulating (6) is necessary when L is a general complete residuated lattice.

 Example 3.19. Take L = [0, 1]. Let us have a family {Mi | i ∈ N} of L-algebras Mi = Mi , ≈Mi , ∅ such that i . That is, a1 ≈M1 b1 = 31 , a2 ≈M2 b2 = 25 , a3 ≈M3 b3 = 37 , . . . . Clearly, Mi = {ai , bi }, and ai ≈Mi bi = 2i+1 hN, ≤i is a directed index set, the universe sets Mi (i ∈ N) are pairwise disjoint, and {hij : Mi → Mj | i ≤ j}, where hij (ai ) = aj , and hij (bi ) = bj is a family of morphisms satisfying (4) and (5). Altogether, hN, ≤i with {Mi | i ∈ N}, and {hij : Mi → Mj | i ≤ j} is a weak direct family. On the other hand, it is not a direct family, because for i, k ∈ N with i ≤ k we have hik (ai ) ≈Mk hik (bi ) < hi,k+1 (ai ) ≈Mk+1 hi,k+1 (bi ). Moreover, we have W θ∞ (ai , bi ) = k≥i hik (ai ) ≈Mk hik (bi ) = 21 ,
 S lim Mi i.e. = ∅), it readily follows that i∈N Mi /θ∞ = [ai ]θ∞ , [bi ]θ∞ . Since lim Mi is of the empty type (F T (X)×T (X) T (X) = X. Thus, for X = {x, y}, and R ∈ L , where R(x, y) = R(y, x) = 12 , and R(x, x) = R(y, y) = 1, we have θ(R) is finitely presented. Now let h : T(X)/θ(R) → lim Mi be defined

= R. Therefore, T(X)/θ(R) by h [x]θ(R) = [aj ]θ∞ , h [y]θ(R) = [bj ]θ∞ . Clearly, h is an ≈-morphism and thus a morphism. Suppose h = g ◦ hk , where g : T (X)/θ(R) → Mk and
Mi is a morphism of the limit cone of lim Mi .
hk : Mk → lim In such a case, h = g ◦ hk yields g [x]θ(R) = ak , g [y]θ(R) = bk . Thus, g cannot be an ≈-morphism since 1 k 2 6≤ 2k+1 . Let us stress a consequential property of the generalized direct limits. If L is infinite and {Mi | i ∈ I} is a weak direct family which is not a direct family, there can be elements a ∈ Mi , b ∈ Mj the homomorphic images of which are distinct in every Mk for k ≥ i, j. However, it can happen that θ∞ (a, b) = 1, i.e. [a]θ∞ = [b]θ∞ due to the general suprema used in (9). Such a situation is apparently ill at least from the standpoint of compatibility with the ordinary direct limits. Indeed, the skeleton ske(lim Mi ) (i.e. an ordinary algebra being the functional part of lim Mi ) is then not isomorphic to the ordinary direct limit of skeletons ske(Mi ). On the other hand, such a situation cannot occur for direct families of L-algebras. Theorem 3.20. Let {Mi | i ∈ I} be a direct family. Then ske(lim Mi ) ∼ = lim{ske(Mi ) | i ∈ I}. Proof. The claim is almost evident. It suffices to show that S for a
∈ Mi , and b ∈ Mj we have θ∞ (a, b) = 1 iff there exists k ≥ i, j such that hik (a) = hjk (b) since then i∈I Mi /θ∞ coincides with its classical counterpart. So, assume that θ∞ (a, b) = 1. Since {Mi | i ∈ I} is a direct family, there is some index k ≥ i, j such that hik (a) ≈Mk hjk (b) = 1. That is, hik (a) = hjk (b). The converse implication holds trivially. Altogether,

D. LIM. AND R. PROD. OF ALG. WITH FUZZY EQUALITIES

11

ske(lim Mi ) ∼ = lim ske(Mi ) since the corresponding functions on ske(lim Mi ) and lim ske(Mi ) are defined the same way.  Example 3.21. Let us consider L = [0, 1] as the structure of truth degrees. We can take a weak direct family from (b) of Remark 3.2. It is evident that lim Mi is a trivial L-algebra but there is not any j ∈ I such that hij (ai ) = hij (bi ). On the other hand, lim ske(Mi ) is a two-element (ordinary) algebra. Hence, Theorem 3.20 is not true for general weak direct families of L-algebras. Remark 3.22. (a) Let us mention an alternative way to generalize direct limits. In the ordinary case [1, 22], a direct limit is sometimes defined to be a factorization of a special subalgebra of a direct product. The direct limit of L-algebras can be approached analogously. Recall that we have already generalized all the necessary notions [5]. Namely, for a directed index set hI, ≤i, and a family {hij : Mi → Mj | i ≤ j} of morphisms satisfying (4) and (5) we can define a set M / by  Q M / = a ∈ i∈I M i | there is i ∈ I such that for j, k ∈ I with i ≤ j ≤ k we have hjk (a(j)) = a(k) . Q Described verbally, M / represents a subset of i∈I M i every element of which respects hij ’s. Furthermore, we W V define a binary L-relation θ/ on M / by θ/ (a, b) = i∈I k≥i a(k) ≈Mk b(k) for every a, b ∈ M / . It can be shown Q that (i) ∅ = 6 M / is a subuniverse of i∈I Mi , and θ/ is a congruence; (ii) for every weak direct family {Mi | i ∈ I} we have lim Mi ∼ = M/ /θ/ ; (iii) for every directed index set hI, ≤i and a family {hij : Mi → Mj | i ≤ j} of morphisms satisfying (4) and (5) there is a weak direct family {gij : Ni → Nj | i ≤ j} such that lim Ni ∼ = M/ /θ/ . The proof is left to the reader. (b) Direct limits in context of particular structures associated with fuzzy sets were studied in [16]. The paper describes direct limits of join spaces which are associated with direct families of fuzzy sets. Hence, [16] deals with classical direct limits of classical structures which are somehow related to collections of fuzzy sets. 4. Reduced products We define reduced products of L-algebras by means of previously defined constructions [5] in much the same way as in the ordinary case. Later on, we introduce a special property called safeness and show its relationship to the essential property (6) of direct families. TheQkey issue of generalizing reduced products to fuzzy setting is how to define a suitable congruence relation on i∈I Mi with respect to a given filter F over I. Recall that in the classical case we put ha, bi ∈ θF

iff

{i ∈ I | a(i) = b(i)} ∈ F.

Thus, on the verbal level: “ha, bi ∈ θF iff the set of indices on which a equals b is large (i.e. belongs to a filter F ).” In what follows, we will proceed in two steps. First, we try to generalize the notion of “being equal on indices from X ∈ F ”. Then, using such a graded equality with respect to some index set, we define an L-relation Q representing for every a, b ∈ i∈I M i a degree to which a equals b over a large set of indices. In the sequel, we use an ordinary filter. That is, we do not fuzzify the notion of a filter itself. We denote a filter by F , and the elements of F will be denoted X, Y, Z, . . . (there is no danger of confusion with the symbol of a type of an L-algebra and with sets of variables, because we use a fixed type and we do not use variables and terms anymore). The rest of this section is devoted to reduced products. Ultraproducts are not discussed. Definition 4.1. Let {Mi | i ∈ I} be a family of L-algebras and let F be a filter over I. Then for every Q a, b ∈ i∈I M i and X ∈ F we define the truth degree [[a ≈ b]]X ∈ L by V [[a ≈ b]]X = i∈X a(i) ≈Mi b(i). Lemma 4.2. Let {Mi | i ∈ I} be a family of L-algebras and let F be a filter over I. Then (i) for Q every X, Y ∈ F , such that X ⊆ Y we have [[a ≈ b]]Y ≤ [[a ≈ b]]X , i∈I Mi b ≤ [[a ≈ b]] (ii) a ≈ W W X for every X ∈ F , (iii) X∈F [[a ≈ b]]X = X1 ,...,Xn ∈F [[a ≈ b]]X1 ∩···∩Xn . Proof. (i) follows directly by DefinitionQ4.1. (ii): Since X ⊆ I ∈ F , (i) yields a ≈ i∈I Mi b = [[a ≈ b]]I ≤ [[a ≈ b]]X . (iii): The “≤”-part follows easily since for each X ∈ F we have X = X ∩· · ·∩X. Conversely, if X1 , . . . , Xn ∈ F then X1 ∩ · · · ∩ Xn ∈ F since every filter is closed under finite intersections. Hence, the “≥”-part is also evident.  Q Now we use [[a ≈ b]]X to define a suitable L-relation on i∈I M i .

12

´ VILEM VYCHODIL

Definition 4.3. Q Let {Mi | i ∈ I} be a family of L-algebras and let F be a filter over I. We define the binary L-relation θF on i∈I M i by W θF (a, b) = X∈F [[a ≈ b]]X Q for all a, b ∈ i∈I M i . Remark 4.4. On the verbal level, [[a ≈ b]]X expresses the truth degree to which it is true that a is equal to b over all indices taken from X. Since X ∈ F are thought of as large subsets, θF (a, b) can be understood as the degree to which “there is a large X such that a equals b over all indices from X”. Lemma 4.5. Let {Mi | i ∈ I} be a family of L-algebras and let F be a filter over I. Then θF is a congruence. Q

Proof. By Lemma 4.2 it readily follows that ≈ i∈I Mi ⊆ θF . Moreover, reflexivity and symmetry of θF follow directly by reflexivity and symmetry of every ≈Mi , respectively. Thus, Q Q it suffices to check transitivity and compatibility with functions of i∈I Mi . Using Lemma 4.2, for a, b, c ∈ i∈I M i we have
W W W θF (a, b) ⊗ θF (b, c) = X∈F [[a ≈ b]]X ⊗ Y ∈F [[b ≈ c]]Y = X,Y ∈F [[a ≈ b]]X ⊗ [[b ≈ c]]Y =
W V V = X,Y ∈F i∈X a(i) ≈Mi b(i) ⊗ j∈Y b(j) ≈Mj c(j) ≤
W V ≤ X,Y ∈F i,j∈X∩Y a(i) ≈Mi b(i) ⊗ b(j) ≈Mj c(j) ≤
W V ≤ X,Y ∈F i∈X∩Y a(i) ≈Mi b(i) ⊗ b(i) ≈Mi c(i) ≤ W V W W ≤ X,Y ∈F i∈X∩Y a(i) ≈Mi c(i) = X,Y ∈F [[a ≈ c]]X∩Y = X∈F [[a ≈ c]]X = θF (a, c). Q Q Thus, θF is transitive. Now we check the compatibility. Take an n-ary f i∈I Mi , and a1 , b1 , . . . , an , bn ∈ i∈I M i . Applying the compatibility of all ≈Mi ’s, we have W Nn V θF (a1 , b1 ) ⊗ · · · ⊗ θF (an , bn ) = X1 ,...,Xn ∈F i=1 ji ∈Xi a(ji ) ≈Mji b(ji ) ≤ Q  Q  W ≤ X1 ,...,Xn ∈F f i∈I Mi (a1 , . . . , an ) ≈ f i∈I Mi (b1 , . . . , bn ) X1 ∩···∩Xn = Q Q Q  Q 
W = X∈F f i∈I Mi (a1 , . . . , an ) ≈ f i∈I Mi (b1 , . . . , bn ) X = θF f i∈I Mi (a1 , . . . , an ), f i∈I Mi (b1 , . . . , bn ) . Q Altogether, θF is a congruence on i∈I Mi .  Finally, we introduce the reduced product of L-algebras.
Q Definition Q 4.6. Let {Mi | i ∈ I} be a family of L-algebras and let F be a filter over I. Then i∈I Mi /θF denoted by F Mi is called the reduced product of {Mi | i ∈ I} modulo F . Q Remark 4.7. Clearly, θF and the corresponding F Mi are determined by {Mi | i ∈ I} and the filter F over I. In borderline cases, θF behaves the same way Q as in the ordinary Q case. Indeed, when F is an improper filter (i.e. ∅ ∈ F ), we have θF (a, b) = 1 for all a, b ∈ i∈I M i . Thus, F Mi is a trivial (one-element) L-algebra. If F is a trivial filter (i.e. F is a proper filter and {i0 } ∈ F for i0 ∈ Q I) it follows that θF = ≈Mi0 for certain i0 ∈ I. Q Q Q ∼ That is, F Mi = Mi0 . Finally, if F = {I} then clearly θF = ≈ i∈I Mi , i.e. F Mi ∼ = i∈I Mi . Q In the ordinary case, the reduced product F Mi is isomorphic to a special direct limit. In the subsequent development, we present an analogous characterization for fuzzy setting. The reduced product of L-algebras will be characterized as a direct limit of certain weak direct family of L-algebras. Let {Mi | i ∈ I} be a family of L-algebras and let us have a filter F over I. For every X ∈ F we can Q consider Q Q Q MX i∈X Mi , the direct product i∈X Mi . For brevity, let MX denote M . That is, M = M , ≈ = ≈ i X i i∈X i∈X Q for an n-ary function symbol f let f MX denote f i∈X Mi . It readily follows that V a ≈MX b = i∈X a(i) ≈Mi b(i) = [[a ≈ b]]X . In addition to that, F can be partially ordered using the ordinary set inclusion. Namely, hF, ⊇i can be thought of as a (downward) directed index set. Clearly, MX , MY are disjoint for each distinct X, Y ∈ F . For X ⊇ Y we define a morphism hXY : MX → MY by hXY (a)(i) = a(i) for every a ∈ MX and i ∈ Y . It is easily seen that {hXY : MX → MY | X ⊇ Y } satisfies conditions (4) and (5). As a consequence, hF, ⊇i, {MX | X ∈ F }, and morphisms hXY (X ⊇ Y ) form a weak direct family. In the sequel, we use the following technical lemma.

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Lemma 4.8. Let h : M → N be a morphism and let φ be a congruence on M such that φ ⊆ θh . Then h = hφ ◦g, where g : M/φ → N is a uniquely determined morphism. Proof. The assertion follows by morphism theorems [5] using analogous arguments as in the ordinary case.  Q ∼ lim MX . Theorem 4.9. Let {Mi | i ∈ I} be a family of L-algebras and let F be a filter over I. Then F Mi = Q Proof. We present a family {hX : MX Q → F Mi } of morphisms Q satisfying DLP with respect to {MX | X ∈ F }. Recall thatQI ∈ F , and MI stands for i∈I Mi . Thus, hIX : i∈I Mi → MX (X ∈ F ) are surjective morphisms. For a, b ∈ i∈I M i we have V θhIX (a, b) = hIX (a) ≈MX hIX (b) = i∈X hIX (a)(i) ≈Mi hIX (b)(i) = V = i∈X a(i) ≈Mi b(i) = [[a ≈ b]]X ≤ θF (a, b). Q Q Therefore, θhIX ⊆ θF (X ∈ F ). Let hθF : i∈I Mi → F Mi denoteQthe natural morphism. By Lemma 4.8, for every X ∈ F there is a uniquely determined morphism hX : MX → F Mi satisfying hθF = hIX ◦ hX . Moreover, we can apply (5) to obtain hθF = hIY ◦ hY = hIX ◦ hXY ◦ hY , i.e. hIX ◦ hXY ◦ hY = hIX ◦ hX . Thus, the surjectivity of hIX yields hX = hXY ◦ hY . Let us have a family {gX : MX → N | X ∈ F } of morphisms Q satisfying gX = hXY ◦ gY (X ⊇ Y ). It remains to show that there is a uniquely determined morphism g : F Mi → N such that gX = hX ◦ g (X ∈ F ). First, Q for a, b ∈ i∈I M i it follows that W W V θF (a, b) = X∈F [[a ≈ b]]X = X∈F i∈X a(i) ≈Mi b(i) = W V W = X∈F i∈X hIX (a)(i) ≈Mi hIX (b)(i) = X∈F hIX (a) ≈MX hIX (b) ≤ W W ≤ X∈F gX (hIX (a)) ≈N gX (hIX (b)) = X∈F gI (a) ≈N gI (b) = gI (a) ≈N gI (b) = θgI (a, b). Q Hence, θF ⊆ θgI . By Lemma 4.8, there is a uniquely determined morphism g : F Mi → N, where gI = hθF ◦ g. Finally, gI = hθF ◦ g = hIX ◦ hX ◦ g and gI =QhIX ◦ gX by the assumption. As a consequence, gQ X = hX ◦ g due to the surjectivity of hIX . Thus, {hX : MX → F Mi } satisfies DLP w.r.t. {MX | X ∈ F }. Now F Mi ∼ = lim MX is a consequence of Theorem 3.14.  As we have seen, for {Mi | i ∈ I} and a filter F over I, {MX | X ∈ F } is a weak direct family of L-algebras. Suppose {MX | X ∈ F } is a direct family of L-algebras and let us look whether the essential property (6) has a natural translation in terms of the properties of F . Definition 4.10. Let {Mi | iQ∈ I} be a family of L-algebras. A filter F over I is called safe with respect to {Mi | i ∈ I} if for every a, b ∈ i∈I M i there is X ∈ F such that θF (a, b) = [[a ≈ b]]X . If K is a class of L-algebras and F is safe w.r.t. every (I-indexed) family of L-algebras taken from K then F is called K-safe. If F is K-safe for arbitrary class K of L-algebras then F is said to be safe. If F is safe with respect to a family {Mi | i ∈ I} Q then F Mi is called the safe reduced product of {Mi | i ∈ I} modulo F . Remark 4.11. Safeness of a filter F with respect to {Mi | i ∈ I} is a nontrivial property. (a) If F = {I} then F is safe. Also everyQtrivial and improper filter is safe. (b) If θF (a, b) is compact for all a, b ∈ i∈I M i then F is safe w.r.t. {Mi | i ∈ I}. Indeed, for any a, b ∈ Q Wn i∈I M i there are X1 , . . . , Xn ∈ F such that θF (a, b) = i=1 [[a ≈ b]]Xi . Since X1 ∩ · · · ∩ Xn ∈ F , it follows that θF (a, b) ≤ [[a ≈ b]]X1 ∩···∩Xn . The converse inequality holds trivially. Thus, if each a ∈ L is compact (see [7]) then every filter F is safe. (c) Take L = [0, 1] as the structure of truth degrees. Let us have an index set N and a family {Mi | i ∈ N} of L-algebras of the empty type, where Mi = {a, b} and a ≈Mi b = 1 − 1i (i ∈ N). Thus, a ≈M1 b = 0, a ≈M2 b = 21 , Q a ≈M3 b = 23 , etc. Let F be Fr´echet filter over N. Take a0 , b0 ∈ i∈N Mi , where a0 (i) = a and b0 (i) = b for all i ∈ N. Clearly, we have θF (a0 , b0 ) = 1 but [[a0 ≈ b0 ]]X < 1 for every X ∈ F . Hence, F is not safe w.r.t. {Mi | i ∈ N}. Lemma 4.12. Let {Mi | i ∈ I} be a family of L-algebras and let F be a filter over I. Then for a ∈ MX , b ∈ MY and any Z ∈ F such that X, Y ⊇ Z we have hXZ (a) ≈MZ hYZ (b) = [[a0 ≈ b0 ]]Z , where a0 , b0 ∈

Q

i∈I M i

satisfy hIX (a0 ) = a and hIY (b0 ) = b.

´ VILEM VYCHODIL

14

Proof. Clearly, we have hXZ (a) ≈MZ hYZ (b) =

V

=

V

i∈Z

hXZ (hIX (a0 ))(i) ≈Mi hYZ (hIY (b0 ))(i) =

i∈Z

a0 (i) ≈Mi b0 (i) = [[a0 ≈ b0 ]]Z ,

i∈Z

hXZ (a)(i) ≈Mi hYZ (b)(i) =

V

i∈Z

hIZ (a0 )(i) ≈Mi hIZ (b0 )(i) =

V

which is the desired equality.



Theorem 4.13. Filter F is safe w.r.t. {Mi | i ∈ I} iff {MX | X ∈ F } is a direct family. Proof. “⇒”: Let F be safe w.r.t. {Mi | i ∈ I}. Take a ∈ MX , b ∈ MY . We have to show that there is Z ∈ F such that X, Y ⊇ Z and hXZ (a) ≈MZ hYZ (b) = hXZ 0 (a) ≈MZ 0 hYZ 0 (b) Q for every Z 0 ∈ F with Z ⊇ Z 0 . Let us have a0 , b0 ∈ i∈I M i such that hIX (a0 ) = a and hIY (b0 ) = b. Since F is safe, we have θF (a0 , b0 ) = [[a0 ≈ b0 ]]Z0 for certain Z0 ∈ F . Put Z = Z0 ∩ X ∩ Y . Clearly, for every Z 0 ∈ F such that Z ⊇ Z 0 , it follows that θF (a0 , b0 ) = [[a0 ≈ b0 ]]Z0 = [[a0 ≈ b0 ]]Z ≥ [[a0 ≈ b0 ]]Z 0 . Moreover, Lemma 4.2 (i) yields [[a0 ≈ b0 ]]Z ≤ [[a0 ≈ b0 ]]Z 0 . Altogether, using Lemma 4.12, we obtain hXZ (a) ≈MZ hYZ (b) = [[a0 ≈ b0 ]]Z = [[a0 ≈ b0 ]]Z 0 = hXZ 0 (a) ≈MZ 0 hYZ 0 (b). Hence, {MX | X ∈ F } is a direct family. Q “⇐”: Let {MX | X ∈ F } be a direct family. For a, b ∈ i∈I M i there is some Z ∈ F such that [[a ≈ b]]Z = hIZ (a) ≈MZ hIZ (b) = hIZ 0 (a) ≈MZ 0 hIZ 0 (b) = [[a ≈ b]]Z 0 holds for every Z 0 ∈ F , Z ⊇ Z 0 . Thus, we have W W W θF (a, b) = X∈F [[a ≈ b]]X ≤ X∈F [[a ≈ b]]X∩Z = X∈F [[a ≈ b]]Z = [[a ≈ b]]Z . The converse inequality follows by the definition of θF . That is, filter F is safe w.r.t. {Mi | i ∈ I}.



Remark 4.14. (a) The construction of a safe reduced product is compatible with its ordinary counterpart in the sense of preserving skeletons: Q Q ske( F Mi ) ∼ = ske(lim MX ) ∼ = lim{ske(MX ) | X ∈ F } ∼ = F ske(Mi ). This is an immediate consequence of Q Theorem 3.20, Q Theorem 4.9, and Theorem 4.13. On the contrary, one can use Remark 4.11 to observe that ske( F Mi ) ∼ = F ske(Mi ) does not hold for general reduced products. (b) In [6] we used fuzzy sets of generalized implications between identities, so-called Horn clauses with truthweighted premises, to characterize sur-reflective classes and semivarieties of L-algebras. In [4], R. Bˇelohl´ avek used fuzzy sets of identities to characterize varieties of L-algebras. All these results pass for any complete residuated lattice as a structure of truth degrees. In case of quasivarieties, however, finiteness of L was used to ensure that each weak direct family is a direct family, and each filter is safe. We also showed that it is not possible to work with unrestricted weak direct families and arbitrary filters because Horn classes of L-algebras are not closed under arbitrary direct limits and reduced products in general. Even if we develop quasivarieties with safe reduced products without invoking a connection to direct limits, Theorem 4.9 and Theorem 4.13 show that the notion of safeness corresponds to the essential property (6) of direct families. An open problem is whether there are other reasonable generalizations of reduced products and direct limits which lead to characterization of quasivarieties over wider subclasses of residuated lattices (hints can be found in [19]). Acknowledgement. Supported by grant no. B1137301 of the Grant Agency of the Academy of Sciences of Czech Republic and by institutional support, research plan MSM 6198959214. References [1] Abramsky S., Gabbay D. M., Maibaum T. S. E.: Handbook of Logic in Computer Science. Volume 1, Oxford University Press, 1992. [2] Bˇ elohl´ avek R.: Fuzzy equational logic. Arch. Math. Logic 41(2002), 83–90. [3] Bˇ elohl´ avek R.: Fuzzy Relational Systems: Foundations and Principles. Kluwer Academic/Plenum Publishers, New York, 2002. [4] Bˇ elohl´ avek R.: Birkhoff variety theorem and fuzzy logic. Arch. Math. Logic 42(2003), 781–790. [5] Bˇ elohl´ avek R., Vychodil V.: Algebras with fuzzy equalities. Fuzzy Sets and Systems 157(2)(2006), 161–201. [6] Bˇ elohl´ avek R., Vychodil V.: Fuzzy Horn logic II: implicationally defined classes. Arch. Math. Logic 45(2)(2006), 149–177. [7] C˘ alug˘ areanu G.: Lattice Concepts of Module Theory. Kluwer, Dordrecht, 2000. [8] Chang C. C., Keisler H. J.: Continuous Model Theory. Princeton University Press, Princeton, NJ, 1966. [9] Esteva F., Godo L.: Monoidal t-norm based logic: towards a logic for left-continuous t-norms. Fuzzy Sets and Systems 124(2001), 271–288. [10] Gerla G.: Fuzzy Logic. Mathematical Tools for Approximate Reasoning. Kluwer, Dordrecht, 2001.

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´ University, Tomkova 40, CZ-779 00, Olomouc, Czech Republic Dept. Computer Science, Palacky E-mail address: [email protected]