Closure Lattices - Semantic Scholar
Closure Lattices John L. Pfaltz University of Virginia
y
October 4, 1994
Abstract
Closure spaces have been previously investigated by Paul Edelman and Robert Jamison as \convex geometries". Consequently, a number of the results given here duplicate theirs. However, we employ a slightly di erent, but equivalent, de ning axiom which gives a new avor to our presentation. The major contribution is the de nition of a partial order on all subsets, not just closed (or convex) subsets. It is shown that the subsets of a closure space, so ordered, form a lattice with regular, though non-modular, properties. Investigation of this lattice becomes our primary focus.
1 Introduction
We let U denote some universe of interest, that is a set of elements, points, or phenomena. Individual points of U will be denoted by lower case letters: a; b; :::; p; q; ::: 2 U. Elements of the power set, 2U , we will denote by upper case letters: :::; X; Y; Z U (or 2 2U ). Our goal will be to partially order these power set elements. A straightforward partial order by inclusion yields a relatively uninteresting boolean lattice, Bn . If, instead, one looks at some underlying structure of the points in U, then uses this to determine the partial order, more interesting results can be obtained. In [18] the author de ned a convexity concept in directed graphs and demonstrated that the collection of convex subsets, partially ordered by inclusion, formed a lower semi-modular lattice. Edelman [6] independently demonstrated the more general result that any lattice of closed sets would be lower semi-modular if the closure operator satis ed an anti-exchange property. He and Jamison re ned these ideas to develop a theory of convex geometries [7]. y
Research supported in part by DOE grant DE-FG05-88ER25063. Written while on leave at the University of Wisconsin-Madison.
1
The relationship between convex geometries, anti-matroids and matroids is well covered by Korte, Lovasz and Schrader in [15]. In all these cited works, the partial order on the power set is by subset inclusion and it is only the lattice of closed subsets that has interesting structure. Adachi, in [1], proposed a di erent partial order which explicitly involved a closure operator.1 But, the power set so ordered is only a semi-lattice. In this paper we introduce a partial ordering of the subsets of 2U induced by any closure operator '. It will be shown that if ' is \uniquely generated" then 2U , so ordered, is a lattice L whose sublattice of closed elements is precisely that of Edelman. Moreover it will be shown that the structure of L over non-closed elements has a regularity that permits the enumeration and reconstruction of uniquely generated closures on n elements.
2 Closure Operators
' U By a closure operator, ', we mean an operator 2U ?! 2 satisfying the standard closure axioms: C1: X X:' C2: X Y implies X:' Y:' C3: X:':' = X:'2 = X:' which are commonly called the Kuratowski Closure Axioms.2 A set X U is said to be closed if X:' = X . The pair (U; ') is called a closure space [10]. Closure operators are common in mathematics and other disciplines. For example, the spanning operator of linear algebra is a closure operator, as are reachability operators in graph theory, and all convex hull operators. In computer science, the transaction operator of concurrent processing is a closure operator as are certain greedy algorithms. The following lemma reviews a number of closure properties that are virtually immediate from the axioms C1, C2, and C3. Lemma 2.1 The following are basic closure properties: (a) If C is closed and X C X:' then C = X:'. (i.e. X:' is the smallest closed set containing X.)
Adachi developed his paper with respect to only a single \lower ideal" closure operator, but it can be easily extended to any uniquely generated closure operator. 2 Note that we are using the standard algebraic notation found in [9] [12], in which binary operations are denoted by in x expressions and unary operations are denoted by sux expressions. This simpli es notation when closure is composed with other operators. The, technically redundant, dot delimiter facilitates automatic parsing in the kind of computer applications for which this theory is being developed. 1
2
(b) X:' [ Y:' (X [ Y ):'. (c) (X \ Y ):' X:' \ Y:'. (d) X:' \ Y:' is closed. (e) X:' \ Y:' = ; for any X; Y implies ;:' = ;. (f) U:' = U. It is well known (c.f. [7] [15]) that T a family F of closed sets satisfying (d) and (f) with a closure operator de ned X:' = Y 2F X Y is an equivalent axiomatization of closure. Several authors also choose to make ;:' = ; an axiom. Given an arbitrary closure operator, ', we de ne an ordering, ' , of 2U by X ' Y if and only if Y \ X:' X Y:' (1) This somewhat unusual de nition is central to our development. For example, Adachi's de nition was X Y if Y \ X:' X Y (omitting the second closure); thereby generating a semi-lattice structure. One can easily show that: Theorem 2.2 ' is a partial order relation on 2U. Proof: Re exivity and anti symmetry are virtual corollaries of the de nition X ' Y Y \ X:' X Y:' and the closure axioms C1-C3. Transitivity is derived from X ' Y and Y ' Z by manipulating the four equivalent containments to yield Z \ X:' Y \ X:' X . The idempotency of ' plus X Y:' and Y Z:' yields X Z:'. 2
Because the ordering on 2U is not simple subset inclusion we must be careful with ordering relationships; we cannot, for example, assume that X \ Y ' X . The following lemma relates properties of X; Y , and Z as sets to their relative order with respect to ' . Lemma 2.3 Let X; Y; Z U and let ' be a closure operator on U. (a) X Y Z and X ' Z; imply X ' Y (b) X Y Z and Z ' X; imply Z ' Y and Y ' X (c) X ' Y and X ' Z imply X ' Y [ Z (d) X ' Z and Y ' Z imply X \ Y ' Z (e) X ' Y implies X ' X [ Y ' Y (f) X ' Y implies X \ Y ' Y (g) X ' Y ' Z implies X \ Z Y . All of the preceding results are based solely on the closure axioms C1, C2, C3, and the de nition of ' in (1). To continue, we must restrict our closure operators somewhat. In addition to the three required closure axioms, we might consider any, or all, of the following properties. 3
C4: if p; q 62 X:' then q 2 (X [ fpg):' implies p 2 (X [ fq g):' C5: X:' = Y:' implies (X \ Y ):' = X:' = Y:' C6: (X \ Y ):' = X:' \ Y:' C7: (X [ Y ):' = X:' [ Y:' The rst of these properties, C4, is called the Steinitz-MacLane exchange property. It characterizes the development of linear algebras, projective geometries, and matroids in terms of closure concepts [3] [10]. One can also postulate an anti-exchange property, of the form if p; q 62 X:' then p 2 (X [ fq g):' implies q 62 (X [ fpg):' (2) which characterizes alignments [8], convex geometries [7], and anti-matroids [15]. In the following development, we will make exclusive use of C5, which we choose to call the unique generation property. However, as asserted in [11] [7], Theorem 2.4 A closure operator is uniquely generated if and only if it satis es the antiexchange property (2). Proof: (Unique generation implies anti-exchange) Let p; q 62 X:', and let p 2 (X [fqg):'. Assume q 2 (X [ fpg):'. Then (X [ fpg):' = (X [ fqg):', so that by the unique generation property X:' = (X [ fpg):' implying p 2 X:', a contradiction. (Anti-exchange implies unique generation) Let X:' = Y:'. Let MX be a minimal set contained in X such that MX :' = X:'. We claim that MX Y . Let p 2 MX . We note that (MX ? fpg):' X:'. Now, suppose p 62 Y . Let ; M 0 Y be a minimal set such that (MX ? fpg [ M 0 ):' = X:'. Let q 2 M 0 and let Z = MX ? fpg [ M 0 ? fqg. Then (MX ? fpg [ M 0 ? fqg):' = Z:' X:'. Now, p; q 62 Z , but p 2 (Z [ fqg):' = X:' and q 2 (Z [ fpg):' = X:' contradicting the anti-exchange axiom. Consequently MX Y . Since MX X \ Y implies MX :' = X:' (X \ Y ):', equality holds. 2
Consequently, uniquely generated closures are completely equivalent to those of abstract convex geometries. Nevertheless, approaching this material from a di erent direction leads to di erent insights that appear to be of value in both lattice theory and computer applications. An example of the latter is the transaction concept of operating systems [17] which permeates concurrent database theory [2]; it is an explicit closure operator. The last two properties are relatively strong. We would note that (a) C6 clearly implies C5; that (b) Kuratowski [16] originally included C7 as one of the closure axioms, because all closed sets in a topological space satisfy it; and that (c) if a closure operator, ', satis es both C6 and C7 then it must be an \identity" operator.3 3
More accurately, ' must have the form X:' = X [ S , where S is any xed subset. If S = ;, so that then for all X , X:' = X , the \identity" operator.
;:' = ;,
4
Z is said to cover X , which we denote X ' Z , if X 6= Z and for any Y such that X ' Y ' Z , either X = Y or Y = Z . Covering relationships are fundamental to the de nition of both modularity and height functions in lattices.
Theorem 2.5 (Fundamental Covering Theorem) If p 62 X then (a) X ' X [ fpg if and only if p 62 X:' (b) X [ fpg' X if and only if p 2 X:' where (a) is a cover if and only if (X [ fpg):' X:' [ fpg and (b) is always a covering relationship.
Moreover, if ' is uniquely generated then (a) and (b) characterize all covering relations in (2U; ' ).
Proof:
(a) Readily X (X [ fpg):'; thus X ' X [ fpg i (X [ fpg) \ X:' X i p 62 X:'. The issue is to establish the covering relationship. Let (X [ fpg):' X:' [ fpg and let Y be such that X ' Y ' X [ fpg. By lemma 2.3(g), X Y . We assume that X 6= Y , else we are done. For q 2 Y ? X; q 62 X:' since X ' Y . Y (X [ fpg):' X:' [ fpg by assumption. Thus, if q 2 Y ? X; q 2 fpg that is q = p. Hence X is covered by X [ fpg in ' . (b) Readily X \ (X [ fpg):' X ; thus X [ fpg X:' i p 2 X:'. Let Y be such that X [fpg'Y ' X . Again X Y . Assume X 6= Y . Let q 2 Y ? X . Y ' X implies Y X:' so in particular q 2 X:'. X [ fpg' Y implies Y \ (X [ fpg):' X [ fpg. So q 2 X [ fpg Thus q = p. Now assume that ' is uniquely generated, and that Y covers X. By lemma 2.3(e), we know X ' X [ Y ' Y , and thus either X = X [ Y or Y = X [ Y by the covering property. Simplifying, either Y X or X Y . In the rst case, suppose 9p 2 X ? Y . Let Z = Y [ fpg so that Y Z X . Since X ' Y , by lemma 2.3(b), X ' Z ' Y . Thus by the covering assumption, Z = X = Y [ fpg Consequently, case (b) of the proposition holds immediately. For the case X Y , assume that jY ? X j 2. Our goal is to show that Y can not cover X. First, suppose that for some p; q 2 Y ? X , (X [ fpg):' = (X [ fqg):'. By unique generation property, (X [ fpg):' = (X [ fqg):' = X:'. Let Z = X [ fpg. X X [ fpg Y . By lemma 2.3(a), X ' X [ fpg, and X [ fpg' Y , since Y \ (X [ fpg):' = Y \ X:' X X [ fpg Y:'. Thus contradicting the covering assumption. On the other hand, if we suppose that (X [ fpg):' 6= (X [ fqg):' for all p; q, then by the pigeon hole principle, for at least one p, (X [ fpg):' X:' [ fpg. Now apply case (a) to X X [ fpg Y
5
to establish ' and contradict the initial covering assumption. Hence, if Y covers X, jY j = jX j 1. 2
As direct corollaries of this fundamental theorem, we have:
Corollary 2.6 Let Z be closed. (a) If Y ' Z , then Y is closed and Y Z . (b) If Y1 ' Z and Y2 ' Z , then there exists X such that X ' Y1 and X ' Y2. Proof:
(a) Y ' Z and Z closed imply Z \ ' (Y ) Y Z , so Y:' X and X X . (b) By (a) above, Y1 and Y2 are closed and contained in Z . Let Y1 = Z ? fpg; Y2 = Z ? fqg, where p 6= q. By theorem 2.5(a), p 62 Y1:'; q 62 Y2 :'. Let X = Y1 ? fqg = Y2 ? fpg = Z ? fp; qg. (We must still show that X ' Y1 and X ' Y2.) Suppose q 2 X:', but since X Y2 , we have q 2 Y2 :' contradicting observation above. So q 62 X:' and similarly p 62 X:'. So X ' Y1 and X ' Y2, and since (X [ fqg):' = Y1 :' X:' [ fqg, these are covers. 2
Corollary 2.7 Let Y ' Z . (a) If Z = Y [ fpg, then both Y and Z are closed. (b) If Z = Y ? fpg, then Z is not closed. Although, we will not establish that (2U; ' ) is a lattice until the following section, we would observe that the interval [;; U] consisting of those subsets Y ' U are precisely the closed subsets of U, by corollary 2.6(a). Moreover they constitute a lower semi-modular sublattice as asserted by 2.6(b) and Thm 3.3 in [6].
3 Generators and Lattices
Let Z be any set closed with respect to '. By a generator of Z , denoted Z:gen', we mean a minimal set Y such that Y:' = Z . With a slight abuse of notation, we shall use the expression Y:gen' with arbitrary Y , with the understanding that if Y is not closed, this means Y:':gen' . Moreover, we will normally omit the subscript '. Readily, if ' satis es the C5 closure property, then the generators of closed sets are unique. (Because, if Y1 and Y2 are distinct minimal sets such that Y1 :' = Z = Y2 :', 6
then (Y1 \ Y2 ):' = Z contradicting minimality.) This is the reason we call it the unique generation property. Clearly we have the equivalent de nition
\
Z:gen' = fYi UjYi:' = Z:'g: i
(3)
Lemma 3.1 If ' is uniquely generated, and if Z 6= ; is closed, (a) p 2 Z:gen if and only if Z ? fpg is closed, in which case Z:gen ? fpg (Z ? fpg):gen; (b) p; q 2 Z:gen implies there exist closed sets Yp ; Yq Z such that p 2 Yp ; q 2 Yq and p 62 Yq ; q 62 Yp ; (c) if ;:' = ;, there exists p 2 Z such that fpg is closed. Proof:
(a) Let p 2 Z:gen. If (Z ? fpg):' = Z , then ((Z ? fpg) \ Z:gen):' = Z contradicting minimality of Z:gen. And if p 62 Z:gen, but Z ? fpg is closed, we have Z:gen:' Z ? fpg 6= Z:', also a contradiction. Let Y be a generator for Z ? fpg, so Z ? fpg is the smallest closed set containing Y . (Y [ fpg):' = Z:' = Z is the smallest closed set containing both Y and p. Hence, by unique generation and minimality of Z:gen, Z:gen Y [ fpg, or Z:gen ? fpg (Z ? fpg):gen. (b) follows directly from (a). Let Yp = Z ? fqg; Yq = Z ? fpg. (c) follows from (a) using induction on jZ j. The condition is necessary to ensure that Z:gen 6= ;. 2 In light of the preceding lemma, those points p 2 Z:gen could be called the extreme points of Z, with the set Z:gen itself called the minimal spanning set [7] or basis [15] of Z . We
prefer the term \generator" because it has fewer other associations. Lemma 3.2 If ' is uniquely generated, then (a) X Y , implies X \ Y:gen X:gen. (b) (X [ Y ):gen X:gen [ Y:gen. (c) X:gen \ Y:gen (X \ Y ):gen.
Proof:
(a) Let p 2 X \ Y:gen. By lemma 3.1, Y:' ? fpg is closed, and (Y ? fpg):gen = Y:gen ? fpg. Suppose p 62 X:gen, then X:gen:' [ (Y:gen ? fpg):' = X:' [ Y:' ? fpg = Y:' (X:gen [ (Y ? fpg):gen):'.
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And, (X:gen [ (Y ?fpg):gen):' Y:'. Thus by unique generation property, ((X:gen [ Y:gen ? fpg) \ Y:gen):' = Y:', contradicting the minimality of Y:gen. (b) Let p 2 (X [ Y ):gen implying (X [ Y ):' ? fpg is closed, and that (X [ Y ? fpg):gen = (X [ Y ):gen ? fpg. We claim that p 2 X [ Y , else (X [ Y ):' ? fpg is the smallest closed set containing X and Y , a contradiction. Assume p 2 X , but p 62 X:gen. One contradicts the minimality of (X [ Y ):gen with an argument virtually identical to (b) above. (c) Similar. 2
Finally, to characterize those sets of elements Yi with the same closure, and generator, in terms of the induced order ' , we have Lemma 3.3 Suppose Y is not closed, and that BY denotes the poset fYi jY:''Yi 'Y:':geng, with induced order ' . Then BY = Bn (boolean algebra on n elements), where n = jY:'j ? jY:':genj, and X Y in Bn if and only if Y X . Proof: By corollary 2.7, no set in BY , except Y:', is closed; Yi:' = Y:'; and all covering relationships Yi ' Yk are of the form Yk = Yi ? fpg. Consequently, 8p 2 Yi ? Y:':gen; Yi ' Yi ? fqg. (If p 2 Y:':gen then (Yi ? fpg):' = 6 Y:' by the de nition of gen.) Hence, BY consists of all subsets of Y:' containing Y:':gen, ordered by inverse inclusion, . 2 Or equivalently, any interval [Y:'; Y:gen] in (2U ; ' ) is a boolean algebra. If we con ne our attention to just closed sets Z , it is easy to show that the height (cardinality of a maximal irreducible chain) is jZ j, as shown in Thm 2.2 [7]. We, however, want a height function for all subsets in 2U, for which the above will be a special case. Theorem 3.4 Let ht(Y ) denote the length n of a maximal irreducible chain Y0 ' Y1 ' Y2 ' ::: 'Yn = Y then ht(Y ) = 2 jY:'j ? jY j ? jY0 j, and ht is a grading of (2U; ' ). In particular, if Y0 = ;:' = ;, then ht(Y ) = 2 jY:'j ? jY j.
Proof: We prove the special case because it is the more important and because the extra machinery needed for the general case tends to obscure the proof structure, even though it is easy to add. We run an induction on ht(Y ). Let ht(Y ) = 0, implying Y = ;; Y:' = ;, and 2 j;j ? j;j = 0. Let ht(Y ) = 1, so ; ' Y . By theorem 2.5, Y = fpg and (; [ fpg):' ;:' [ fpg or fpg:' fpg (establishing that only closed singleton sets cover ;) so jY:'j = jY j = 1, and ht(Y ) = 2 jY:'j?jY j = 1. Assume the induction hypothesis is true for 8X such that ht(X ) < n, and let ht(Y ) = n. 8
1) Y is closed: By the corollary 2.6(a) 8X ' Y , X is closed, Y = X [ fpg, and ht(Y ) = 2 jY:'j ? jY j = 2 (jX:'j + 1) ? (jX j + 1) = 2 jY:'j ? jX j + 1 = ht(X ) + 1. 2) Y is not closed: Observe that by corollary 2.7, we must have Y = X ? fpg (or X = Y [ fpg), p 2 Y:', and jY j = jX j ? 1. 2a) X:' = Y:': In this case, X 2 BY the boolean algebra of lemma 3.3, and ht(X ) + 1 = 2 jX:'j ? jX j + 1 = 2 jY:'j ? jY j = ht(Y ). 2b) X:' 6= Y:': Since X = Y [ fpg and p 2 Y:', X:' 6= Y:' implies p 2 Y:':gen. By lemma 3.1(a), X:' [fpg is closed, and so X:' [fpg = Y:' implying jY:'j = jX:'j +1. Consequently, ht(X ) + 1 = 2 jX:'j ? jX j + 1 = 2 (jY:'j ? 1) ? (jY j + 1) + 1 = 2 jY:'j ? jY j. 2
As noted earlier, Edelman [6], Edelman and Jamison [7], and the author [18] have shown that the closed sets of U, partially ordered by inclusion, form a lattice and have discussed it some detail. We have observed that the partial ordering on 2U developed by Adachi [1] yields only a semi-lattice. A major result of this paper is the demonstration that for any uniquely generated closure, ', the partial ordering of 2U de ned by (1) is in fact a lattice.
Theorem 3.5 If ' is uniquely generated then (2U; ') is a lattice with \ [ \ inf (X1; :::; Xn) = [ (Xi :') \ ( Xi)] [ ( (Xi:')):gen i
Proof: Let I = [(\iXi :') \ ([iXi )] [ (\iXi:'):gen.
i
i
We claim I:' = \iXi :' because: (a) (\iXi :'):gen:' = \iXi :', and (b) the latter intersection of closed sets is closed, so that, (\iXi :'):' = \iXi :'. From (b) we have that (\i Xi :'):' \ ([i Xi )) (\i Xi :'):' \ ([i Xi ):' = \i Xi :' \ ([i Xi ):'
\i Xi :':
which combined with (a) yields I:' \iXi :'. On the other hand, because (\i Xi :'):gen I we have \iXi :' I:'. We must show that 8k; I ' Xk (or Xk \ I:' I Xk :'). The rst containment follows from
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Xk \ I:' = Xk \ (\i Xi :') [k Xk \ (\iXi :') I . The second containment is immediate, because \iXi :' \ ([iXi ) Xi :' and (\i XI :'):gen \iXi :' Xi :'.
Suppose that for all k; Y ' Xk , then Y ' I because (a) Xk \ Y:' Y implies [[k Xk \ \k Xk :'] \ Y:' Y . And, since Y Xk :'; 8k implies Y:' \i Xi :', we have by lemma3.2(a), Y:'\(\iXi :'):gen Y:':gen Y . So, I \ Y:' Y . (b) Y Xk :'; 8k implies Y \k Xk :' = I:'. Having demonstrated that the inf operator exists, we need only establish the existence of a maximal element. We claim it is U:gen. Let X U. Again by lemma 3.2(a), X:' \ U:gen (X:'):gen X . And, readily X U:gen:' = U, so X ' U:gen. 2 Note that the dominant term of this inf operator, commonly denoted by ^, is \i Xi :', as
one might expect. In the next section, we will give examples which require its restriction to [i Xi to keep the inf within the original set, and require its augmentation to include the generators, (\i Xi :'):gen. Finally, we observe that X \ Y X ^ Y . The lattice (2U; ' ) (as described in theorem 3.5) we call the closure lattice induced by ', or more simply the closure lattice.
4 Examples
In this section, we examine two representative closure lattices. First, let U = fa; b; c; dg. The 16 subsets of 2U and their closures have been listed in the following table. X X.' X X.' X X.' X X.'
; ;
fdg fabcdg fbcg fabcg fabdg fabcdg
fag fag fabg fabg fbdg fabcdg facdg fabcdg
fbg fabg facg facg fcdg
fabcdg fbcdg fabcdg
fcg facg fadg fabcdg fabcg fabcg fabcdg fabcdg
Table 1: A closure, ', on U = fa; b; c; dg The reader can verify that ' so de ned on this small set really is a closure operator, and that it is uniquely generated. The resulting closure lattice, L, is shown in Figure 1. This gure illustrates several of the results of the preceding sections. The interval [;; fabcdg] consists precisely of the closed subsets of U, and is lower semi-modular as required by corollary 2.6. (In this case it is actually distributive.) This sublattice has been drawn with 10
d
bcd bc b
abcd
c
abc
ab
ac
a 0
Figure 1: Closure Lattice, given closure ' above solid lines for emphasis. However, the entire lattice is not lower semi-modular, because fbcg covers both fbg and fcg, but neither covers fbg^ fcg = fag. Nor is it upper semi-modular. The subsets fbg; fcg; fbcg, and fdg are generators for the closed sets fabg; facg; fabcg, and fabcdg respectively; while fag is its own generator. Except for the element fbcdg, the boolean algebra comprising the interval [fabcdg; fdg] (lemma 3.3) has been only schematically indicated as an ellipse to avoid useless clutter. Observe, that in fbg ^ fcg, the term [(fbg:' \ fcg:') \ (fbg [ fcg) = ;. So, in this case, fbg ^ fcg = (fbg:' \ fcg:'):gen, as required in theorem 3.5. The preceding closure operator was de ned ex cathedra. More often they are derived from some underlying relationships or properties of U. On any given universe U of n points there are a wealth of distinct closure operators, as we will show in the next section. If U is a partially ordered set, then there are at least 3 natural closure operators corresponding to left ideals, right ideals, and convex intervals.4 For example, if one de nes a left ideal closure Y:' = fxj(9y 2 Y )[x y]g on the 7 point graph of Figure 2 one obtains the somewhat more complex closure lattice of Figure 3 which we will use to motivate the results of the following section. Here again, the sublattice of closed sets, or interval [;; U], has been indicated by solid lines, while the boolean algebras, Bn = [X:'; X:gen], have been denoted by dotted \ellipses". Only a few of the covering relationships between \adjacent" closure-generator intervals have been shown. 4 Many would call these \upper" and \lower" ideals, but when the base universe is ordered, the author orients it from left to right for illustrative purposes in order to minimize confusion with the closure lattice order, which is oriented top down.
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a d
f
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c
Figure 2: An acyclic graph ? g
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def cef
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abcdefg
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abcdef
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abcde
abcdf
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ade cde
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Figure 3: Closure lattice associated with ? The compression of a closure lattice with 27 = 128 elements into just 17 closure-generator pairs facilitates an ecient computer representation of closure spaces and their lattices. In light of (1) and (3), a more compressed representation of just the closed sets is also sucient; but for many applications it is computationally more expensive. These are not the only closures on 7 points. Clearly the arbitrary distribution of n 12
points in a Euclidean d-space gives rise to many di erent convex geometries, e.g. consider the convex sets in 2-space determined by the spatial position of the 7 points in Figure 2. And convex geometries may be generated by undirected graphs with appropriate properties, e.g. block [14] or geodesic graphs, [8]. Processes can also give rise to uniquely generated closures. Both [15] and [5] enumerate various shelling processes that give rise to matroids which satisfy the exchange property, anti-matroids which satisfy the unique generation property, and greedoids (from greedy algorithms) that generalize both. Some applications have both graph and shelling aspects. For example, one may regard the universe as consisting of the set E of edges of a directed graph, rather than its points or vertices. Then one can recursively de ne the transitive closure of E on P by E:' = f(x; z)j(9y 6= x 2 P )[(x; y) 2 E; (y; z) 2 E:']g This corresponds to the customary transitive closure, or path, relation. After verifying the three basic closure axioms, one shows
Lemma 4.1 If E:' is a partial order of P then ' is uniquely generated. Proof: Let E1 :' = E2:' and let them be a partial order on P so that E jP 0 :' = E:'jP 0 where 0
P P is also a partial order. The lemma is easily veri ed for any E on small, nite sets P, so inductively assume it is true for all P0 P. Readily, (E1 \ E2):' E1:', so we need only show the other containment to establish unique generation. Let (x; z ) 2 E1:' implying 9y1 2 P; y1 6= x such that (x; y1 ) 2 E1; (y1 ; z ) 2 E1:'. If (x; y1) 2 E2 we will be done because, rst (x; y1) 2 E1 \ E2 and, second E:' a partial order implies E:'jP ?fxg = E jP ?fxg :', so (y1 ; z ) 2 E1:' implies (y1 ; z ) 2 (E1 \ E2 ):'. These together imply (x; z ) 2 (E1 \ E2):'. But (x; y1) need not be an element of E2. Then (x; y1 ) 2 E1 implies (x; y1) 2 E2:' (since E1:' = E2:'). So 9y2 6= x and y2 6= y1 (since E2:' is a partial order) such that (x; y2 ) 2 E2 ; (y2; y1 ) 2 E2:'. If (x; y2) 2 E1 \ E2 , we are done, using the argument above. If not, 9y3 6= x, y3 6= y2 ; y1 such that (x; y3) 2 E1; (y3 ; y2) 2 E1 :', etc. In this manner we generate a descending sequence of points x : : :yi ; yi?1 ; : : :y2 ; y1
where for even i; (yi+1 ; yi ) 2 E1 ; (yi; yi?1 ) 2 E2 . This sequence has a minimal element y, for which (x; y) 2 E1 \ E2 . 2
Now, by lemma 3.1(a) one can delete any edge from E:gen to yield a new poset on P with exactly one less edge. This shelling technique has been employed in [4] to generate sequences of posets with n elements. To see that the condition of Lemma 4.1 cannot be relaxed, consider a directed Peterson graph, as shown in Figure 4 in which E1 is denoted by solid edges, E2 is denoted by dashed 13
a b
c
E1: E2:
d
e
Figure 4: A graph ? which is not uniquely generated by transitive closure edges. Readily, E1:' = E2 :', yet E1 \ E2 = ;. Both E1 and E2 are minimal generators of this cyclic order.
5 Lattice Structure of Non-Closed Subsets
The structure of the closed sets of any U with respect to a uniquely generated closure ' is well known. They constitute a lower semi-modular sublattice of L comprising interval [;; U], in which the partial order is subset inclusion, c.f. [7], or corollaries 2.6, 2.7 of the Fundamental Covering Theorem. These can be restated in terms of X ? fpg rather than X [ fpg. Lemma 5.1 Let ' be uniquely generated (a) X ' X ? fpg if and only if p 2 (X ? fpg):' (b) If X is closed, then X ? fpg ' X if and only if p 62 (X ? fpg):' (c) If X is not closed, then X ? fpg ' X if and only if (X ? fpg):' ' X:'. (d) If X is closed, then either X ' X ? fpg or X ? fpg ' X . We now want to uncover the structural relationships between non-closed elements (sets of U). We know that for any X , the interval [X:'; X:gen] is isomorphic to the boolean algebra Bn (lemma 3.3), but this provides no information regarding the structure between elements in distinct intervals. Our goal is to show that these [closed set, generator] boolean algebras are stacked, in increasing size, with a covering structure that echoes that of the closed sets which constitute their least elements; that is, the shape of the closure lattices shown in Figures 1 and 3 is not accidental. We begin with 14
Lemma 5.2 Let X1 be closed in (U; ') and let Z1 = X1:gen. Let X2 ' X1 (so X2 = X1 ? fpg is also closed) and let Z2 = X2:gen. Then for all Y2 2 [X2; Z2], there exists a unique Y1 2 [X1; Z1] such that Y2 ' Y1 . Moreover, Y1 = Y2 [ fpg, and Y1 = X1 ? where = X2 ? Y2 . Proof: By theorem 2.5, p 62 X2:'. By lemma 3.1(a) p 2 X1 = X1:gen and Z1 ? fpg Z2. Given Y2 2 [X2; Z2 ], let Y1 = Y2 [ fpg. We rst claim that Y1 2 [X1 ; Z1], or equivalently (Y2 [ fpg):' = X1 . Because Z2 Y2 X2 , because X2 [ fpg = X1 , and because Z1 ? fpg Z2 , this follows easily. We next claim that Y2 ' Y1 . Y2 ' Y1 because p 62 Y2:' = X2 :'. Moreover, (X2 [ fpg):' Y2 :' [ fpg = X2 [ fpg = X1 . So by theorem 2.5, it is a cover. Finally, let Y 0 2 [X1 ; Z1]. A corollary to theorem 2.5 is that all covering relationships involve exactly one point, so Y 0 = Y2 [ fqg. If q 6= p then Y 0 2 [X1; Z1 ] implies q 2 X1 , which in turn implies q 2 X2 = Y2 :', thereby contradiction (again by theorem 2.5) that Y2' Y 0 . 2
It can be instructive to ll in some of the missing covering relationships of Figure 3 that are asserted by this lemma. For example, the closed element fabdg is covered by fabcdg and fabdf g (with = fcg and ff g respectively). Consequently, fdg is covered by fcdg (which is shown) and fdf g 2 [abdf; f ] (which is not) respectively. Following are two direct corollaries of this lemma. The rst is virtually trivial. The second, which generalizes the structure between elements in di erent [closed set, generator] intervals is fundamental. We call it the Fundamental Structure Theorem, or FST. Corollary 5.3 Let X 'Y , then [X:'; X:gen] = Bm and [Y:'; Y:gen] = Bn where m n. Theorem 5.4 (Fundamental Structure Theorem) Let X:' ' Y:' and let X 2 [X:'; X:gen]. There exists a unique Y 2 [Y:'; Y:gen] such that X ' Y , where Y is minimal wrt. ' (maximal wrt. ). Moreover Y = X [ where = Y:' ? X:' and Y = Y:' ? where = X:' ? X . The FST, which is shown by a simple induction argument, asserts the existence of sets above any given set in the closure lattice L. For example consider X = fadg 2 [abd; d] in Figure 3. Since fabdg'fabcdeg, this theorem asserts that X ' Y = facf g with = fbg and = fcf g. The unique existence of these elements can be crucial in arguments regarding the continuity of discrete operators, an issue which is not considered in this paper. Because a closure lattice can be regarded as a nested collection of boolean algebras, that are themselves partially ordered by increasing size, it is possible to explicitly characterize all closure operators on n points. Since each of the 2jU j elements of a closure lattice over U belong to some boolean algebra Bk , we have 15
Lemma 5.5 Let ' be any uniquely generated closure operator on U, with n = jUj. Let ak
denote the number of [closed set, generator] intervals isomorphic to Bk , then a0 20 + a1 21 + + an?1 2n?1 + an 2n = 2n: (4) The sequence < a0 ; a1; ; an > of non-negative integers can be regarded as a partition of 2n . We call it the characteristic trace of ' on U. Readily, (a) an 6= 0 if and only if ak = 0 for all k < n, in which case ' is the trivial closure X:' = U for all X U, and U:gen = ;; (b) a0 denotes the number of closed sets whichPare their own generators; (c) a0 6= 0 if and only if ;:' = ;; (d) a0 must be even; and (e) ak denotes the total number of closed subsets of U with respect to '. We observe in passing that (c) and (d) together imply lemma 3.1(c). One can recursively generate all distinct closure traces, because if < a0 ; ; ak?1 ; ak ; ; an > is a characteristic closure trace, then < a0 ; ; 2 ak?1 ; ak ? 1; ; an > is a trace as well. Using a simple program that generates all traces in lexicographic order and counts them, one obtains Table 2. The second column enumerates all closure traces on n points, the
closures with ;:' = ; n all closures 3 10 6 36 26 4 5 202 166 1,828 1,626 6 7 27,338 25,510 692,004 664,666 8 9 30,251,722 29,559,718 Table 2: Enumeration of distinct characteristic traces of n point closures third column those with a0 6= 0, or by the observation above ;:' = ;. (For n > 9, these values exceed the length of a long integer on the computer used to generate the table.) As mentioned earlier, there exist many di erent closures on a space of n points. Given any arbitrary n point closure trace, such as the traces < 20; 2; 2; 0; 0; 0 > or < 8; 0; 4; 3; 1; 0; 1; 0 > (the trace of Figure 3), one can generate actual closure spaces with these characteristics. That is, Theorem 5.6 Let < a0; a1; ; an?1; an > be any sequence of non-negative integers such P n that k=0 ak 2k = 2n . There exists a closure operator ' on U; jUj = n, for which trace(') =< a0 ; ; an >.
Proof: This is most easily demonstrated by a procedure which actually generates the closure
operator ', or more precisely a collection of [closed set, generator] pairs which de nes the operator.
16
In our implementation, points are lower case letters, a; b; c; : : :; z and point sets are lexicographically ordered strings of distinct points. Therefore the function rst points is well de ned; it returns a point set consisting of the rst j points according to this arbitrary order. generate (int n, int a[ ], point set U)
Given n and a[0], ... a[n], generate the closure pairs of the corresponding closure over U.
point int set point set list
p i, j, k closed, closure def new cl, new gen, old cl, old gen queue
closed = empty set k = n if a[k] = 0 decrement k until a[k] != 0 new cl = U new gen = first points (j, new cl ) insert new cl into closed add (new cl, new gen) to queue insert (new cl, new gen) into closure def while queue is not empty do
f
remove (old cl, old gen) from queue for each p in old gen do
f
if a[k] = 0 decrement k until a[k] != 0 new cl = old cl - fpg if new cl not in closed
f
(old cl, old gen) covers a boolean interval (new cl, new gen) which is isomorphic to B[n].
insert new cl into closed j = size of(new cl) - k new gen = first points (j, new cl) add (new cl, new gen) to queue insert (new cl, new gen) into closure def decrement a[k]
g
g
g
return closure def
To show that the algorithm is correct, one needs only show that for closed X , if p 2 X:gen then X covers X ? fpg. But this follows directly from lemma 5.1(b). 2 Figure 5 illustrates the closure lattice corresponding to the set of closure pairs f[X:'; X:gen]g returned by the generate procedure when given the trace < 20; 2; 2; 0; 0; 0 >. Unfortunately, trace sequences satisfying (4) do not uniquely characterize closure lattices. Consider the lattice of Figure 6(b) which also has the trace < 20; 2; 2; 0; 0; 0 > and compare it with Figure 5. Figure 6(b) is obtained from the graph to its left, using a convex 17
abc
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Figure 5: Closure lattice with trace < 20; 2; 2; 0; 0; 0 > returned by the procedure generate interval closure5 Y:' = fxjy1 x y2 , where y1 ; y2 2 Y g. Consequently, Table 2 only can be regarded as providing a lower bound on the number of distinct closure operators, and closure lattices, on n points. Acknowledgment: The author would like to acknowledge the contribution of Mark Pleszkoch of IBM, Federal Systems Division, who provided early versions of theorems 2.5 and 3.5.
References [1] Takanori Adachi. Powerposets. Information & Control, 66:138{162, 1985. [2] P.A. Bernstein, V. Hadzilacos, and N. Goodman. Concurrency Control and Recovery in Database Systems. Addison-Wesley, 1987. 5
called order convex in [13] and [7].
18
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Figure 6: Closure lattice with trace < 20; 2; 2; 0; 0; 0 > induced by convex interval closure on (a) [3] Garrett Birkho . Lattice Theory. Colloquium Publ., Volume XXV. Amer. Math. Soc., 1940. [4] Richard A. Brualdi, Hyung Chan Jung, and William T. Trotter, Jr. On the poset of all posets on n elements. Discrete Applied Mathematics, 1994. To appear. [5] Brenda L. Dietrich. Matroids and antimatroids | a survey. Discrete Mathematics, 78:223{237, 1989. [6] Paul H. Edelman. Meet-distributive lattices and the anti-exchange closure. Algebra Universalis, 10(3):290{299, 1980. [7] Paul H. Edelman and Robert E. Jamison. The theory of convex geometries. Geometriae Dedicata, 19(3):247{270, Dec. 1985. 19
[8] Martin Farber and Robert E. Jamison. Convexity in graphs and hypergraphs. SIAM J. Algebra and Discrete Methods, 7(3):433{444, July 1986. [9] George Gratzer. Universal Algebra. Van Nostrand, Princeton, NJ, 1968. [10] George Gratzer. General Lattice Theory. Academic Press, 1978. [11] A. J. Ho man. Binding constraints and Helly numbers. In 2nd Intern'l Conf. on Combinatorial Math., volume 319, pages 284{288. Annals of the N.Y. Acad. of Sciences, 1979. [12] Nathan Jacobson. Lectures in Abstract Algebra. Van Nostrand, Princeton, NJ, 1961. [13] Robert E. Jamison-Waldner. A convexity characterization of ordered sets. Congressus Numerantium, 24, part II:529{540, Apr. 1979. [14] Robert E. Jamison-Waldner. Convexity and block graphs. Congressus Numerantium, 33:129{142, Dec. 1981. [15] Bernhard Korte, Laszlo Lovasz, and Rainer Schrader. Greedoids. Springer-Verlag, Berlin, 1991. [16] Kazimierz Kuratowski. Introduction to Set Theory and Topology. Pergamon Press, 1972. [17] B.W. Lampson. Atomic transactions. In B.W. Lampson, editor, Distributed Systems{ Architecture and Implementation, pages 246{265. Springer Verlag, 1981. [18] John L. Pfaltz. Convexity in directed graphs. J. of Comb. Theory, 10(2):143{162, Apr. 1971.
20
y
October 4, 1994
Abstract
Closure spaces have been previously investigated by Paul Edelman and Robert Jamison as \convex geometries". Consequently, a number of the results given here duplicate theirs. However, we employ a slightly di erent, but equivalent, de ning axiom which gives a new avor to our presentation. The major contribution is the de nition of a partial order on all subsets, not just closed (or convex) subsets. It is shown that the subsets of a closure space, so ordered, form a lattice with regular, though non-modular, properties. Investigation of this lattice becomes our primary focus.
1 Introduction
We let U denote some universe of interest, that is a set of elements, points, or phenomena. Individual points of U will be denoted by lower case letters: a; b; :::; p; q; ::: 2 U. Elements of the power set, 2U , we will denote by upper case letters: :::; X; Y; Z U (or 2 2U ). Our goal will be to partially order these power set elements. A straightforward partial order by inclusion yields a relatively uninteresting boolean lattice, Bn . If, instead, one looks at some underlying structure of the points in U, then uses this to determine the partial order, more interesting results can be obtained. In [18] the author de ned a convexity concept in directed graphs and demonstrated that the collection of convex subsets, partially ordered by inclusion, formed a lower semi-modular lattice. Edelman [6] independently demonstrated the more general result that any lattice of closed sets would be lower semi-modular if the closure operator satis ed an anti-exchange property. He and Jamison re ned these ideas to develop a theory of convex geometries [7]. y
Research supported in part by DOE grant DE-FG05-88ER25063. Written while on leave at the University of Wisconsin-Madison.
1
The relationship between convex geometries, anti-matroids and matroids is well covered by Korte, Lovasz and Schrader in [15]. In all these cited works, the partial order on the power set is by subset inclusion and it is only the lattice of closed subsets that has interesting structure. Adachi, in [1], proposed a di erent partial order which explicitly involved a closure operator.1 But, the power set so ordered is only a semi-lattice. In this paper we introduce a partial ordering of the subsets of 2U induced by any closure operator '. It will be shown that if ' is \uniquely generated" then 2U , so ordered, is a lattice L whose sublattice of closed elements is precisely that of Edelman. Moreover it will be shown that the structure of L over non-closed elements has a regularity that permits the enumeration and reconstruction of uniquely generated closures on n elements.
2 Closure Operators
' U By a closure operator, ', we mean an operator 2U ?! 2 satisfying the standard closure axioms: C1: X X:' C2: X Y implies X:' Y:' C3: X:':' = X:'2 = X:' which are commonly called the Kuratowski Closure Axioms.2 A set X U is said to be closed if X:' = X . The pair (U; ') is called a closure space [10]. Closure operators are common in mathematics and other disciplines. For example, the spanning operator of linear algebra is a closure operator, as are reachability operators in graph theory, and all convex hull operators. In computer science, the transaction operator of concurrent processing is a closure operator as are certain greedy algorithms. The following lemma reviews a number of closure properties that are virtually immediate from the axioms C1, C2, and C3. Lemma 2.1 The following are basic closure properties: (a) If C is closed and X C X:' then C = X:'. (i.e. X:' is the smallest closed set containing X.)
Adachi developed his paper with respect to only a single \lower ideal" closure operator, but it can be easily extended to any uniquely generated closure operator. 2 Note that we are using the standard algebraic notation found in [9] [12], in which binary operations are denoted by in x expressions and unary operations are denoted by sux expressions. This simpli es notation when closure is composed with other operators. The, technically redundant, dot delimiter facilitates automatic parsing in the kind of computer applications for which this theory is being developed. 1
2
(b) X:' [ Y:' (X [ Y ):'. (c) (X \ Y ):' X:' \ Y:'. (d) X:' \ Y:' is closed. (e) X:' \ Y:' = ; for any X; Y implies ;:' = ;. (f) U:' = U. It is well known (c.f. [7] [15]) that T a family F of closed sets satisfying (d) and (f) with a closure operator de ned X:' = Y 2F X Y is an equivalent axiomatization of closure. Several authors also choose to make ;:' = ; an axiom. Given an arbitrary closure operator, ', we de ne an ordering, ' , of 2U by X ' Y if and only if Y \ X:' X Y:' (1) This somewhat unusual de nition is central to our development. For example, Adachi's de nition was X Y if Y \ X:' X Y (omitting the second closure); thereby generating a semi-lattice structure. One can easily show that: Theorem 2.2 ' is a partial order relation on 2U. Proof: Re exivity and anti symmetry are virtual corollaries of the de nition X ' Y Y \ X:' X Y:' and the closure axioms C1-C3. Transitivity is derived from X ' Y and Y ' Z by manipulating the four equivalent containments to yield Z \ X:' Y \ X:' X . The idempotency of ' plus X Y:' and Y Z:' yields X Z:'. 2
Because the ordering on 2U is not simple subset inclusion we must be careful with ordering relationships; we cannot, for example, assume that X \ Y ' X . The following lemma relates properties of X; Y , and Z as sets to their relative order with respect to ' . Lemma 2.3 Let X; Y; Z U and let ' be a closure operator on U. (a) X Y Z and X ' Z; imply X ' Y (b) X Y Z and Z ' X; imply Z ' Y and Y ' X (c) X ' Y and X ' Z imply X ' Y [ Z (d) X ' Z and Y ' Z imply X \ Y ' Z (e) X ' Y implies X ' X [ Y ' Y (f) X ' Y implies X \ Y ' Y (g) X ' Y ' Z implies X \ Z Y . All of the preceding results are based solely on the closure axioms C1, C2, C3, and the de nition of ' in (1). To continue, we must restrict our closure operators somewhat. In addition to the three required closure axioms, we might consider any, or all, of the following properties. 3
C4: if p; q 62 X:' then q 2 (X [ fpg):' implies p 2 (X [ fq g):' C5: X:' = Y:' implies (X \ Y ):' = X:' = Y:' C6: (X \ Y ):' = X:' \ Y:' C7: (X [ Y ):' = X:' [ Y:' The rst of these properties, C4, is called the Steinitz-MacLane exchange property. It characterizes the development of linear algebras, projective geometries, and matroids in terms of closure concepts [3] [10]. One can also postulate an anti-exchange property, of the form if p; q 62 X:' then p 2 (X [ fq g):' implies q 62 (X [ fpg):' (2) which characterizes alignments [8], convex geometries [7], and anti-matroids [15]. In the following development, we will make exclusive use of C5, which we choose to call the unique generation property. However, as asserted in [11] [7], Theorem 2.4 A closure operator is uniquely generated if and only if it satis es the antiexchange property (2). Proof: (Unique generation implies anti-exchange) Let p; q 62 X:', and let p 2 (X [fqg):'. Assume q 2 (X [ fpg):'. Then (X [ fpg):' = (X [ fqg):', so that by the unique generation property X:' = (X [ fpg):' implying p 2 X:', a contradiction. (Anti-exchange implies unique generation) Let X:' = Y:'. Let MX be a minimal set contained in X such that MX :' = X:'. We claim that MX Y . Let p 2 MX . We note that (MX ? fpg):' X:'. Now, suppose p 62 Y . Let ; M 0 Y be a minimal set such that (MX ? fpg [ M 0 ):' = X:'. Let q 2 M 0 and let Z = MX ? fpg [ M 0 ? fqg. Then (MX ? fpg [ M 0 ? fqg):' = Z:' X:'. Now, p; q 62 Z , but p 2 (Z [ fqg):' = X:' and q 2 (Z [ fpg):' = X:' contradicting the anti-exchange axiom. Consequently MX Y . Since MX X \ Y implies MX :' = X:' (X \ Y ):', equality holds. 2
Consequently, uniquely generated closures are completely equivalent to those of abstract convex geometries. Nevertheless, approaching this material from a di erent direction leads to di erent insights that appear to be of value in both lattice theory and computer applications. An example of the latter is the transaction concept of operating systems [17] which permeates concurrent database theory [2]; it is an explicit closure operator. The last two properties are relatively strong. We would note that (a) C6 clearly implies C5; that (b) Kuratowski [16] originally included C7 as one of the closure axioms, because all closed sets in a topological space satisfy it; and that (c) if a closure operator, ', satis es both C6 and C7 then it must be an \identity" operator.3 3
More accurately, ' must have the form X:' = X [ S , where S is any xed subset. If S = ;, so that then for all X , X:' = X , the \identity" operator.
;:' = ;,
4
Z is said to cover X , which we denote X ' Z , if X 6= Z and for any Y such that X ' Y ' Z , either X = Y or Y = Z . Covering relationships are fundamental to the de nition of both modularity and height functions in lattices.
Theorem 2.5 (Fundamental Covering Theorem) If p 62 X then (a) X ' X [ fpg if and only if p 62 X:' (b) X [ fpg' X if and only if p 2 X:' where (a) is a cover if and only if (X [ fpg):' X:' [ fpg and (b) is always a covering relationship.
Moreover, if ' is uniquely generated then (a) and (b) characterize all covering relations in (2U; ' ).
Proof:
(a) Readily X (X [ fpg):'; thus X ' X [ fpg i (X [ fpg) \ X:' X i p 62 X:'. The issue is to establish the covering relationship. Let (X [ fpg):' X:' [ fpg and let Y be such that X ' Y ' X [ fpg. By lemma 2.3(g), X Y . We assume that X 6= Y , else we are done. For q 2 Y ? X; q 62 X:' since X ' Y . Y (X [ fpg):' X:' [ fpg by assumption. Thus, if q 2 Y ? X; q 2 fpg that is q = p. Hence X is covered by X [ fpg in ' . (b) Readily X \ (X [ fpg):' X ; thus X [ fpg X:' i p 2 X:'. Let Y be such that X [fpg'Y ' X . Again X Y . Assume X 6= Y . Let q 2 Y ? X . Y ' X implies Y X:' so in particular q 2 X:'. X [ fpg' Y implies Y \ (X [ fpg):' X [ fpg. So q 2 X [ fpg Thus q = p. Now assume that ' is uniquely generated, and that Y covers X. By lemma 2.3(e), we know X ' X [ Y ' Y , and thus either X = X [ Y or Y = X [ Y by the covering property. Simplifying, either Y X or X Y . In the rst case, suppose 9p 2 X ? Y . Let Z = Y [ fpg so that Y Z X . Since X ' Y , by lemma 2.3(b), X ' Z ' Y . Thus by the covering assumption, Z = X = Y [ fpg Consequently, case (b) of the proposition holds immediately. For the case X Y , assume that jY ? X j 2. Our goal is to show that Y can not cover X. First, suppose that for some p; q 2 Y ? X , (X [ fpg):' = (X [ fqg):'. By unique generation property, (X [ fpg):' = (X [ fqg):' = X:'. Let Z = X [ fpg. X X [ fpg Y . By lemma 2.3(a), X ' X [ fpg, and X [ fpg' Y , since Y \ (X [ fpg):' = Y \ X:' X X [ fpg Y:'. Thus contradicting the covering assumption. On the other hand, if we suppose that (X [ fpg):' 6= (X [ fqg):' for all p; q, then by the pigeon hole principle, for at least one p, (X [ fpg):' X:' [ fpg. Now apply case (a) to X X [ fpg Y
5
to establish ' and contradict the initial covering assumption. Hence, if Y covers X, jY j = jX j 1. 2
As direct corollaries of this fundamental theorem, we have:
Corollary 2.6 Let Z be closed. (a) If Y ' Z , then Y is closed and Y Z . (b) If Y1 ' Z and Y2 ' Z , then there exists X such that X ' Y1 and X ' Y2. Proof:
(a) Y ' Z and Z closed imply Z \ ' (Y ) Y Z , so Y:' X and X X . (b) By (a) above, Y1 and Y2 are closed and contained in Z . Let Y1 = Z ? fpg; Y2 = Z ? fqg, where p 6= q. By theorem 2.5(a), p 62 Y1:'; q 62 Y2 :'. Let X = Y1 ? fqg = Y2 ? fpg = Z ? fp; qg. (We must still show that X ' Y1 and X ' Y2.) Suppose q 2 X:', but since X Y2 , we have q 2 Y2 :' contradicting observation above. So q 62 X:' and similarly p 62 X:'. So X ' Y1 and X ' Y2, and since (X [ fqg):' = Y1 :' X:' [ fqg, these are covers. 2
Corollary 2.7 Let Y ' Z . (a) If Z = Y [ fpg, then both Y and Z are closed. (b) If Z = Y ? fpg, then Z is not closed. Although, we will not establish that (2U; ' ) is a lattice until the following section, we would observe that the interval [;; U] consisting of those subsets Y ' U are precisely the closed subsets of U, by corollary 2.6(a). Moreover they constitute a lower semi-modular sublattice as asserted by 2.6(b) and Thm 3.3 in [6].
3 Generators and Lattices
Let Z be any set closed with respect to '. By a generator of Z , denoted Z:gen', we mean a minimal set Y such that Y:' = Z . With a slight abuse of notation, we shall use the expression Y:gen' with arbitrary Y , with the understanding that if Y is not closed, this means Y:':gen' . Moreover, we will normally omit the subscript '. Readily, if ' satis es the C5 closure property, then the generators of closed sets are unique. (Because, if Y1 and Y2 are distinct minimal sets such that Y1 :' = Z = Y2 :', 6
then (Y1 \ Y2 ):' = Z contradicting minimality.) This is the reason we call it the unique generation property. Clearly we have the equivalent de nition
\
Z:gen' = fYi UjYi:' = Z:'g: i
(3)
Lemma 3.1 If ' is uniquely generated, and if Z 6= ; is closed, (a) p 2 Z:gen if and only if Z ? fpg is closed, in which case Z:gen ? fpg (Z ? fpg):gen; (b) p; q 2 Z:gen implies there exist closed sets Yp ; Yq Z such that p 2 Yp ; q 2 Yq and p 62 Yq ; q 62 Yp ; (c) if ;:' = ;, there exists p 2 Z such that fpg is closed. Proof:
(a) Let p 2 Z:gen. If (Z ? fpg):' = Z , then ((Z ? fpg) \ Z:gen):' = Z contradicting minimality of Z:gen. And if p 62 Z:gen, but Z ? fpg is closed, we have Z:gen:' Z ? fpg 6= Z:', also a contradiction. Let Y be a generator for Z ? fpg, so Z ? fpg is the smallest closed set containing Y . (Y [ fpg):' = Z:' = Z is the smallest closed set containing both Y and p. Hence, by unique generation and minimality of Z:gen, Z:gen Y [ fpg, or Z:gen ? fpg (Z ? fpg):gen. (b) follows directly from (a). Let Yp = Z ? fqg; Yq = Z ? fpg. (c) follows from (a) using induction on jZ j. The condition is necessary to ensure that Z:gen 6= ;. 2 In light of the preceding lemma, those points p 2 Z:gen could be called the extreme points of Z, with the set Z:gen itself called the minimal spanning set [7] or basis [15] of Z . We
prefer the term \generator" because it has fewer other associations. Lemma 3.2 If ' is uniquely generated, then (a) X Y , implies X \ Y:gen X:gen. (b) (X [ Y ):gen X:gen [ Y:gen. (c) X:gen \ Y:gen (X \ Y ):gen.
Proof:
(a) Let p 2 X \ Y:gen. By lemma 3.1, Y:' ? fpg is closed, and (Y ? fpg):gen = Y:gen ? fpg. Suppose p 62 X:gen, then X:gen:' [ (Y:gen ? fpg):' = X:' [ Y:' ? fpg = Y:' (X:gen [ (Y ? fpg):gen):'.
7
And, (X:gen [ (Y ?fpg):gen):' Y:'. Thus by unique generation property, ((X:gen [ Y:gen ? fpg) \ Y:gen):' = Y:', contradicting the minimality of Y:gen. (b) Let p 2 (X [ Y ):gen implying (X [ Y ):' ? fpg is closed, and that (X [ Y ? fpg):gen = (X [ Y ):gen ? fpg. We claim that p 2 X [ Y , else (X [ Y ):' ? fpg is the smallest closed set containing X and Y , a contradiction. Assume p 2 X , but p 62 X:gen. One contradicts the minimality of (X [ Y ):gen with an argument virtually identical to (b) above. (c) Similar. 2
Finally, to characterize those sets of elements Yi with the same closure, and generator, in terms of the induced order ' , we have Lemma 3.3 Suppose Y is not closed, and that BY denotes the poset fYi jY:''Yi 'Y:':geng, with induced order ' . Then BY = Bn (boolean algebra on n elements), where n = jY:'j ? jY:':genj, and X Y in Bn if and only if Y X . Proof: By corollary 2.7, no set in BY , except Y:', is closed; Yi:' = Y:'; and all covering relationships Yi ' Yk are of the form Yk = Yi ? fpg. Consequently, 8p 2 Yi ? Y:':gen; Yi ' Yi ? fqg. (If p 2 Y:':gen then (Yi ? fpg):' = 6 Y:' by the de nition of gen.) Hence, BY consists of all subsets of Y:' containing Y:':gen, ordered by inverse inclusion, . 2 Or equivalently, any interval [Y:'; Y:gen] in (2U ; ' ) is a boolean algebra. If we con ne our attention to just closed sets Z , it is easy to show that the height (cardinality of a maximal irreducible chain) is jZ j, as shown in Thm 2.2 [7]. We, however, want a height function for all subsets in 2U, for which the above will be a special case. Theorem 3.4 Let ht(Y ) denote the length n of a maximal irreducible chain Y0 ' Y1 ' Y2 ' ::: 'Yn = Y then ht(Y ) = 2 jY:'j ? jY j ? jY0 j, and ht is a grading of (2U; ' ). In particular, if Y0 = ;:' = ;, then ht(Y ) = 2 jY:'j ? jY j.
Proof: We prove the special case because it is the more important and because the extra machinery needed for the general case tends to obscure the proof structure, even though it is easy to add. We run an induction on ht(Y ). Let ht(Y ) = 0, implying Y = ;; Y:' = ;, and 2 j;j ? j;j = 0. Let ht(Y ) = 1, so ; ' Y . By theorem 2.5, Y = fpg and (; [ fpg):' ;:' [ fpg or fpg:' fpg (establishing that only closed singleton sets cover ;) so jY:'j = jY j = 1, and ht(Y ) = 2 jY:'j?jY j = 1. Assume the induction hypothesis is true for 8X such that ht(X ) < n, and let ht(Y ) = n. 8
1) Y is closed: By the corollary 2.6(a) 8X ' Y , X is closed, Y = X [ fpg, and ht(Y ) = 2 jY:'j ? jY j = 2 (jX:'j + 1) ? (jX j + 1) = 2 jY:'j ? jX j + 1 = ht(X ) + 1. 2) Y is not closed: Observe that by corollary 2.7, we must have Y = X ? fpg (or X = Y [ fpg), p 2 Y:', and jY j = jX j ? 1. 2a) X:' = Y:': In this case, X 2 BY the boolean algebra of lemma 3.3, and ht(X ) + 1 = 2 jX:'j ? jX j + 1 = 2 jY:'j ? jY j = ht(Y ). 2b) X:' 6= Y:': Since X = Y [ fpg and p 2 Y:', X:' 6= Y:' implies p 2 Y:':gen. By lemma 3.1(a), X:' [fpg is closed, and so X:' [fpg = Y:' implying jY:'j = jX:'j +1. Consequently, ht(X ) + 1 = 2 jX:'j ? jX j + 1 = 2 (jY:'j ? 1) ? (jY j + 1) + 1 = 2 jY:'j ? jY j. 2
As noted earlier, Edelman [6], Edelman and Jamison [7], and the author [18] have shown that the closed sets of U, partially ordered by inclusion, form a lattice and have discussed it some detail. We have observed that the partial ordering on 2U developed by Adachi [1] yields only a semi-lattice. A major result of this paper is the demonstration that for any uniquely generated closure, ', the partial ordering of 2U de ned by (1) is in fact a lattice.
Theorem 3.5 If ' is uniquely generated then (2U; ') is a lattice with \ [ \ inf (X1; :::; Xn) = [ (Xi :') \ ( Xi)] [ ( (Xi:')):gen i
Proof: Let I = [(\iXi :') \ ([iXi )] [ (\iXi:'):gen.
i
i
We claim I:' = \iXi :' because: (a) (\iXi :'):gen:' = \iXi :', and (b) the latter intersection of closed sets is closed, so that, (\iXi :'):' = \iXi :'. From (b) we have that (\i Xi :'):' \ ([i Xi )) (\i Xi :'):' \ ([i Xi ):' = \i Xi :' \ ([i Xi ):'
\i Xi :':
which combined with (a) yields I:' \iXi :'. On the other hand, because (\i Xi :'):gen I we have \iXi :' I:'. We must show that 8k; I ' Xk (or Xk \ I:' I Xk :'). The rst containment follows from
9
Xk \ I:' = Xk \ (\i Xi :') [k Xk \ (\iXi :') I . The second containment is immediate, because \iXi :' \ ([iXi ) Xi :' and (\i XI :'):gen \iXi :' Xi :'.
Suppose that for all k; Y ' Xk , then Y ' I because (a) Xk \ Y:' Y implies [[k Xk \ \k Xk :'] \ Y:' Y . And, since Y Xk :'; 8k implies Y:' \i Xi :', we have by lemma3.2(a), Y:'\(\iXi :'):gen Y:':gen Y . So, I \ Y:' Y . (b) Y Xk :'; 8k implies Y \k Xk :' = I:'. Having demonstrated that the inf operator exists, we need only establish the existence of a maximal element. We claim it is U:gen. Let X U. Again by lemma 3.2(a), X:' \ U:gen (X:'):gen X . And, readily X U:gen:' = U, so X ' U:gen. 2 Note that the dominant term of this inf operator, commonly denoted by ^, is \i Xi :', as
one might expect. In the next section, we will give examples which require its restriction to [i Xi to keep the inf within the original set, and require its augmentation to include the generators, (\i Xi :'):gen. Finally, we observe that X \ Y X ^ Y . The lattice (2U; ' ) (as described in theorem 3.5) we call the closure lattice induced by ', or more simply the closure lattice.
4 Examples
In this section, we examine two representative closure lattices. First, let U = fa; b; c; dg. The 16 subsets of 2U and their closures have been listed in the following table. X X.' X X.' X X.' X X.'
; ;
fdg fabcdg fbcg fabcg fabdg fabcdg
fag fag fabg fabg fbdg fabcdg facdg fabcdg
fbg fabg facg facg fcdg
fabcdg fbcdg fabcdg
fcg facg fadg fabcdg fabcg fabcg fabcdg fabcdg
Table 1: A closure, ', on U = fa; b; c; dg The reader can verify that ' so de ned on this small set really is a closure operator, and that it is uniquely generated. The resulting closure lattice, L, is shown in Figure 1. This gure illustrates several of the results of the preceding sections. The interval [;; fabcdg] consists precisely of the closed subsets of U, and is lower semi-modular as required by corollary 2.6. (In this case it is actually distributive.) This sublattice has been drawn with 10
d
bcd bc b
abcd
c
abc
ab
ac
a 0
Figure 1: Closure Lattice, given closure ' above solid lines for emphasis. However, the entire lattice is not lower semi-modular, because fbcg covers both fbg and fcg, but neither covers fbg^ fcg = fag. Nor is it upper semi-modular. The subsets fbg; fcg; fbcg, and fdg are generators for the closed sets fabg; facg; fabcg, and fabcdg respectively; while fag is its own generator. Except for the element fbcdg, the boolean algebra comprising the interval [fabcdg; fdg] (lemma 3.3) has been only schematically indicated as an ellipse to avoid useless clutter. Observe, that in fbg ^ fcg, the term [(fbg:' \ fcg:') \ (fbg [ fcg) = ;. So, in this case, fbg ^ fcg = (fbg:' \ fcg:'):gen, as required in theorem 3.5. The preceding closure operator was de ned ex cathedra. More often they are derived from some underlying relationships or properties of U. On any given universe U of n points there are a wealth of distinct closure operators, as we will show in the next section. If U is a partially ordered set, then there are at least 3 natural closure operators corresponding to left ideals, right ideals, and convex intervals.4 For example, if one de nes a left ideal closure Y:' = fxj(9y 2 Y )[x y]g on the 7 point graph of Figure 2 one obtains the somewhat more complex closure lattice of Figure 3 which we will use to motivate the results of the following section. Here again, the sublattice of closed sets, or interval [;; U], has been indicated by solid lines, while the boolean algebras, Bn = [X:'; X:gen], have been denoted by dotted \ellipses". Only a few of the covering relationships between \adjacent" closure-generator intervals have been shown. 4 Many would call these \upper" and \lower" ideals, but when the base universe is ordered, the author orients it from left to right for illustrative purposes in order to minimize confusion with the closure lattice order, which is oriented top down.
11
a d
f
b
g e
c
Figure 2: An acyclic graph ? g
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ef
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def cef
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de
abcdefg
7
abcdef
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abc
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3
bc
ac
ab
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c
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1
ade cde
ae
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ht
Figure 3: Closure lattice associated with ? The compression of a closure lattice with 27 = 128 elements into just 17 closure-generator pairs facilitates an ecient computer representation of closure spaces and their lattices. In light of (1) and (3), a more compressed representation of just the closed sets is also sucient; but for many applications it is computationally more expensive. These are not the only closures on 7 points. Clearly the arbitrary distribution of n 12
points in a Euclidean d-space gives rise to many di erent convex geometries, e.g. consider the convex sets in 2-space determined by the spatial position of the 7 points in Figure 2. And convex geometries may be generated by undirected graphs with appropriate properties, e.g. block [14] or geodesic graphs, [8]. Processes can also give rise to uniquely generated closures. Both [15] and [5] enumerate various shelling processes that give rise to matroids which satisfy the exchange property, anti-matroids which satisfy the unique generation property, and greedoids (from greedy algorithms) that generalize both. Some applications have both graph and shelling aspects. For example, one may regard the universe as consisting of the set E of edges of a directed graph, rather than its points or vertices. Then one can recursively de ne the transitive closure of E on P by E:' = f(x; z)j(9y 6= x 2 P )[(x; y) 2 E; (y; z) 2 E:']g This corresponds to the customary transitive closure, or path, relation. After verifying the three basic closure axioms, one shows
Lemma 4.1 If E:' is a partial order of P then ' is uniquely generated. Proof: Let E1 :' = E2:' and let them be a partial order on P so that E jP 0 :' = E:'jP 0 where 0
P P is also a partial order. The lemma is easily veri ed for any E on small, nite sets P, so inductively assume it is true for all P0 P. Readily, (E1 \ E2):' E1:', so we need only show the other containment to establish unique generation. Let (x; z ) 2 E1:' implying 9y1 2 P; y1 6= x such that (x; y1 ) 2 E1; (y1 ; z ) 2 E1:'. If (x; y1) 2 E2 we will be done because, rst (x; y1) 2 E1 \ E2 and, second E:' a partial order implies E:'jP ?fxg = E jP ?fxg :', so (y1 ; z ) 2 E1:' implies (y1 ; z ) 2 (E1 \ E2 ):'. These together imply (x; z ) 2 (E1 \ E2):'. But (x; y1) need not be an element of E2. Then (x; y1 ) 2 E1 implies (x; y1) 2 E2:' (since E1:' = E2:'). So 9y2 6= x and y2 6= y1 (since E2:' is a partial order) such that (x; y2 ) 2 E2 ; (y2; y1 ) 2 E2:'. If (x; y2) 2 E1 \ E2 , we are done, using the argument above. If not, 9y3 6= x, y3 6= y2 ; y1 such that (x; y3) 2 E1; (y3 ; y2) 2 E1 :', etc. In this manner we generate a descending sequence of points x : : :yi ; yi?1 ; : : :y2 ; y1
where for even i; (yi+1 ; yi ) 2 E1 ; (yi; yi?1 ) 2 E2 . This sequence has a minimal element y, for which (x; y) 2 E1 \ E2 . 2
Now, by lemma 3.1(a) one can delete any edge from E:gen to yield a new poset on P with exactly one less edge. This shelling technique has been employed in [4] to generate sequences of posets with n elements. To see that the condition of Lemma 4.1 cannot be relaxed, consider a directed Peterson graph, as shown in Figure 4 in which E1 is denoted by solid edges, E2 is denoted by dashed 13
a b
c
E1: E2:
d
e
Figure 4: A graph ? which is not uniquely generated by transitive closure edges. Readily, E1:' = E2 :', yet E1 \ E2 = ;. Both E1 and E2 are minimal generators of this cyclic order.
5 Lattice Structure of Non-Closed Subsets
The structure of the closed sets of any U with respect to a uniquely generated closure ' is well known. They constitute a lower semi-modular sublattice of L comprising interval [;; U], in which the partial order is subset inclusion, c.f. [7], or corollaries 2.6, 2.7 of the Fundamental Covering Theorem. These can be restated in terms of X ? fpg rather than X [ fpg. Lemma 5.1 Let ' be uniquely generated (a) X ' X ? fpg if and only if p 2 (X ? fpg):' (b) If X is closed, then X ? fpg ' X if and only if p 62 (X ? fpg):' (c) If X is not closed, then X ? fpg ' X if and only if (X ? fpg):' ' X:'. (d) If X is closed, then either X ' X ? fpg or X ? fpg ' X . We now want to uncover the structural relationships between non-closed elements (sets of U). We know that for any X , the interval [X:'; X:gen] is isomorphic to the boolean algebra Bn (lemma 3.3), but this provides no information regarding the structure between elements in distinct intervals. Our goal is to show that these [closed set, generator] boolean algebras are stacked, in increasing size, with a covering structure that echoes that of the closed sets which constitute their least elements; that is, the shape of the closure lattices shown in Figures 1 and 3 is not accidental. We begin with 14
Lemma 5.2 Let X1 be closed in (U; ') and let Z1 = X1:gen. Let X2 ' X1 (so X2 = X1 ? fpg is also closed) and let Z2 = X2:gen. Then for all Y2 2 [X2; Z2], there exists a unique Y1 2 [X1; Z1] such that Y2 ' Y1 . Moreover, Y1 = Y2 [ fpg, and Y1 = X1 ? where = X2 ? Y2 . Proof: By theorem 2.5, p 62 X2:'. By lemma 3.1(a) p 2 X1 = X1:gen and Z1 ? fpg Z2. Given Y2 2 [X2; Z2 ], let Y1 = Y2 [ fpg. We rst claim that Y1 2 [X1 ; Z1], or equivalently (Y2 [ fpg):' = X1 . Because Z2 Y2 X2 , because X2 [ fpg = X1 , and because Z1 ? fpg Z2 , this follows easily. We next claim that Y2 ' Y1 . Y2 ' Y1 because p 62 Y2:' = X2 :'. Moreover, (X2 [ fpg):' Y2 :' [ fpg = X2 [ fpg = X1 . So by theorem 2.5, it is a cover. Finally, let Y 0 2 [X1 ; Z1]. A corollary to theorem 2.5 is that all covering relationships involve exactly one point, so Y 0 = Y2 [ fqg. If q 6= p then Y 0 2 [X1; Z1 ] implies q 2 X1 , which in turn implies q 2 X2 = Y2 :', thereby contradiction (again by theorem 2.5) that Y2' Y 0 . 2
It can be instructive to ll in some of the missing covering relationships of Figure 3 that are asserted by this lemma. For example, the closed element fabdg is covered by fabcdg and fabdf g (with = fcg and ff g respectively). Consequently, fdg is covered by fcdg (which is shown) and fdf g 2 [abdf; f ] (which is not) respectively. Following are two direct corollaries of this lemma. The rst is virtually trivial. The second, which generalizes the structure between elements in di erent [closed set, generator] intervals is fundamental. We call it the Fundamental Structure Theorem, or FST. Corollary 5.3 Let X 'Y , then [X:'; X:gen] = Bm and [Y:'; Y:gen] = Bn where m n. Theorem 5.4 (Fundamental Structure Theorem) Let X:' ' Y:' and let X 2 [X:'; X:gen]. There exists a unique Y 2 [Y:'; Y:gen] such that X ' Y , where Y is minimal wrt. ' (maximal wrt. ). Moreover Y = X [ where = Y:' ? X:' and Y = Y:' ? where = X:' ? X . The FST, which is shown by a simple induction argument, asserts the existence of sets above any given set in the closure lattice L. For example consider X = fadg 2 [abd; d] in Figure 3. Since fabdg'fabcdeg, this theorem asserts that X ' Y = facf g with = fbg and = fcf g. The unique existence of these elements can be crucial in arguments regarding the continuity of discrete operators, an issue which is not considered in this paper. Because a closure lattice can be regarded as a nested collection of boolean algebras, that are themselves partially ordered by increasing size, it is possible to explicitly characterize all closure operators on n points. Since each of the 2jU j elements of a closure lattice over U belong to some boolean algebra Bk , we have 15
Lemma 5.5 Let ' be any uniquely generated closure operator on U, with n = jUj. Let ak
denote the number of [closed set, generator] intervals isomorphic to Bk , then a0 20 + a1 21 + + an?1 2n?1 + an 2n = 2n: (4) The sequence < a0 ; a1; ; an > of non-negative integers can be regarded as a partition of 2n . We call it the characteristic trace of ' on U. Readily, (a) an 6= 0 if and only if ak = 0 for all k < n, in which case ' is the trivial closure X:' = U for all X U, and U:gen = ;; (b) a0 denotes the number of closed sets whichPare their own generators; (c) a0 6= 0 if and only if ;:' = ;; (d) a0 must be even; and (e) ak denotes the total number of closed subsets of U with respect to '. We observe in passing that (c) and (d) together imply lemma 3.1(c). One can recursively generate all distinct closure traces, because if < a0 ; ; ak?1 ; ak ; ; an > is a characteristic closure trace, then < a0 ; ; 2 ak?1 ; ak ? 1; ; an > is a trace as well. Using a simple program that generates all traces in lexicographic order and counts them, one obtains Table 2. The second column enumerates all closure traces on n points, the
closures with ;:' = ; n all closures 3 10 6 36 26 4 5 202 166 1,828 1,626 6 7 27,338 25,510 692,004 664,666 8 9 30,251,722 29,559,718 Table 2: Enumeration of distinct characteristic traces of n point closures third column those with a0 6= 0, or by the observation above ;:' = ;. (For n > 9, these values exceed the length of a long integer on the computer used to generate the table.) As mentioned earlier, there exist many di erent closures on a space of n points. Given any arbitrary n point closure trace, such as the traces < 20; 2; 2; 0; 0; 0 > or < 8; 0; 4; 3; 1; 0; 1; 0 > (the trace of Figure 3), one can generate actual closure spaces with these characteristics. That is, Theorem 5.6 Let < a0; a1; ; an?1; an > be any sequence of non-negative integers such P n that k=0 ak 2k = 2n . There exists a closure operator ' on U; jUj = n, for which trace(') =< a0 ; ; an >.
Proof: This is most easily demonstrated by a procedure which actually generates the closure
operator ', or more precisely a collection of [closed set, generator] pairs which de nes the operator.
16
In our implementation, points are lower case letters, a; b; c; : : :; z and point sets are lexicographically ordered strings of distinct points. Therefore the function rst points is well de ned; it returns a point set consisting of the rst j points according to this arbitrary order. generate (int n, int a[ ], point set U)
Given n and a[0], ... a[n], generate the closure pairs of the corresponding closure over U.
point int set point set list
p i, j, k closed, closure def new cl, new gen, old cl, old gen queue
closed = empty set k = n if a[k] = 0 decrement k until a[k] != 0 new cl = U new gen = first points (j, new cl ) insert new cl into closed add (new cl, new gen) to queue insert (new cl, new gen) into closure def while queue is not empty do
f
remove (old cl, old gen) from queue for each p in old gen do
f
if a[k] = 0 decrement k until a[k] != 0 new cl = old cl - fpg if new cl not in closed
f
(old cl, old gen) covers a boolean interval (new cl, new gen) which is isomorphic to B[n].
insert new cl into closed j = size of(new cl) - k new gen = first points (j, new cl) add (new cl, new gen) to queue insert (new cl, new gen) into closure def decrement a[k]
g
g
g
return closure def
To show that the algorithm is correct, one needs only show that for closed X , if p 2 X:gen then X covers X ? fpg. But this follows directly from lemma 5.1(b). 2 Figure 5 illustrates the closure lattice corresponding to the set of closure pairs f[X:'; X:gen]g returned by the generate procedure when given the trace < 20; 2; 2; 0; 0; 0 >. Unfortunately, trace sequences satisfying (4) do not uniquely characterize closure lattices. Consider the lattice of Figure 6(b) which also has the trace < 20; 2; 2; 0; 0; 0 > and compare it with Figure 5. Figure 6(b) is obtained from the graph to its left, using a convex 17
abc
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Figure 5: Closure lattice with trace < 20; 2; 2; 0; 0; 0 > returned by the procedure generate interval closure5 Y:' = fxjy1 x y2 , where y1 ; y2 2 Y g. Consequently, Table 2 only can be regarded as providing a lower bound on the number of distinct closure operators, and closure lattices, on n points. Acknowledgment: The author would like to acknowledge the contribution of Mark Pleszkoch of IBM, Federal Systems Division, who provided early versions of theorems 2.5 and 3.5.
References [1] Takanori Adachi. Powerposets. Information & Control, 66:138{162, 1985. [2] P.A. Bernstein, V. Hadzilacos, and N. Goodman. Concurrency Control and Recovery in Database Systems. Addison-Wesley, 1987. 5
called order convex in [13] and [7].
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(a)
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O (b)
Figure 6: Closure lattice with trace < 20; 2; 2; 0; 0; 0 > induced by convex interval closure on (a) [3] Garrett Birkho . Lattice Theory. Colloquium Publ., Volume XXV. Amer. Math. Soc., 1940. [4] Richard A. Brualdi, Hyung Chan Jung, and William T. Trotter, Jr. On the poset of all posets on n elements. Discrete Applied Mathematics, 1994. To appear. [5] Brenda L. Dietrich. Matroids and antimatroids | a survey. Discrete Mathematics, 78:223{237, 1989. [6] Paul H. Edelman. Meet-distributive lattices and the anti-exchange closure. Algebra Universalis, 10(3):290{299, 1980. [7] Paul H. Edelman and Robert E. Jamison. The theory of convex geometries. Geometriae Dedicata, 19(3):247{270, Dec. 1985. 19
[8] Martin Farber and Robert E. Jamison. Convexity in graphs and hypergraphs. SIAM J. Algebra and Discrete Methods, 7(3):433{444, July 1986. [9] George Gratzer. Universal Algebra. Van Nostrand, Princeton, NJ, 1968. [10] George Gratzer. General Lattice Theory. Academic Press, 1978. [11] A. J. Ho man. Binding constraints and Helly numbers. In 2nd Intern'l Conf. on Combinatorial Math., volume 319, pages 284{288. Annals of the N.Y. Acad. of Sciences, 1979. [12] Nathan Jacobson. Lectures in Abstract Algebra. Van Nostrand, Princeton, NJ, 1961. [13] Robert E. Jamison-Waldner. A convexity characterization of ordered sets. Congressus Numerantium, 24, part II:529{540, Apr. 1979. [14] Robert E. Jamison-Waldner. Convexity and block graphs. Congressus Numerantium, 33:129{142, Dec. 1981. [15] Bernhard Korte, Laszlo Lovasz, and Rainer Schrader. Greedoids. Springer-Verlag, Berlin, 1991. [16] Kazimierz Kuratowski. Introduction to Set Theory and Topology. Pergamon Press, 1972. [17] B.W. Lampson. Atomic transactions. In B.W. Lampson, editor, Distributed Systems{ Architecture and Implementation, pages 246{265. Springer Verlag, 1981. [18] John L. Pfaltz. Convexity in directed graphs. J. of Comb. Theory, 10(2):143{162, Apr. 1971.
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