John L. Pfaltz. Closure Lattices. Discrete Mathematics, 154:217--236, 1996.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of Systems Science, Graduate School of Arts and Sciences, The University of Tokyo 3 8 1, Komaba, Meguro, Tokyo, 153 8902, Japan kashiwa graco.c.u tokyo.ac. jp Abstract Convex geometries are closure spaces which satisfy anti exchange property, and they are known as the dual of antimatroids [2, 5, 7]. We consider set functions defined on the family of all extreme sets of a convex geometry. Faigle and Kern [3, 4] presented a greedy algorithm of linear programmings for poset shellings, and Kruger [6] introduced a b submodular function on the family of all extreme sets on a poset shelling and ....

J. L. Pfaltz, Closure lattices, Discrete Mathematics 154, 1996, 217--236.

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John L. Pfaltz. Closure Lattices. Discrete Mathematics, 154:217--236, 1996.

....C, X: T XaeY Y 2 C. And conversely, C consists just of those closed sets Z where Z: Z. The key aspect of closure theory for data mining lies in the concept of a minimal generator of a closed set Z, denoted Z:fl. This is a minimal set X (by inclusion) whose closure will be Z. See [24, 26] for background here. Frequent set analysis yields associations of the form ab abcde. We have maximal information content when ab is minimal and abcde is maximal. This will occur if the consequent abcde is closed and its antecedent ab is its generator. This is one way that closure theory has ....

John L. Pfaltz. Closure Lattices. Discrete Mathematics, 154:217--236, 1996.

....be used to automatically convert a formal concept lattice into a system of implications. 1 Overview Matroids and antimatroids can be studied either in terms of a family F of feasible sets and a shelling operator oe [1,11] or in terms of a collection C of closed sets and a closure operator [3,14]. There exists a considerable amount of confusion, and an equally great richness, because these are two distinct approaches to precisely the same concepts. Given an antimatroid universe, U, every feasible set F 2 F is the complement of a closed set U Gamma C; C 2 C, and conversely. In this paper ....

....is a tight relationship between any closed set and its generators, which is expressed by Theorem 1.1 (Fundamental Covering Theorem) Let Z 2 C be any closed set. Z Gamma fpg is closed if and only if p 2 Z:fl. An ordering, X Y iff Y X: X Y: on all subsets X; Y U, was introduced in [14] which also introduced graphic representations such as in Figure 1. When closure is taken to be Y: L = fxj(9y 2 Y ) x y]g, the lattice of Figure 1(b) a b c d abcd abc ab ac a bcd abd acd cd ad bd d bc b c (a) b) Fig. 1. Closure defined on a graph G (a) and its lattice, L, b) ....

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John L. Pfaltz. Closure Lattices. Discrete Mathematics, 154:217--236, 1996.

....to concept analysis are reported in [16] 2 Closure Spaces An operator is a closure operator if X X: X Y ) X: Y: and X: X: The Galois closure on binary relations is one kind of discrete closure operator. A more general treatment of closure spaces has been advanced in [9, 12]. A central idea in these papers is that of the generators of a closed set, Z, denoted Z:fl, by which we mean a minimal set Y such that Y: Z. For example, with a convex hull closure operator, the generators of a convex n gon are its n vertices (or extreme points) 3 An n gon is uniquely ....

....are its n vertices (or extreme points) 3 An n gon is uniquely determined by its generators. Whenever the generators of a closed set must be unique, we say the closure operator is uniquely generated and call the resulting closure space an antimatroid. 4 Much of the closure literature, e.g. [2, 3, 4, 9, 12] assumes antimatroid closure. Using concepts from closure spaces, it is quite straightforward to generate the concept lattice while simultaneously determining the generators of these closed concepts. For example, the single attribute e generates the closed concept acde. That is, feg: R Gamma1 = ....

John L. Pfaltz. Closure Lattices. Discrete Mathematics, 154:217--236, 1996.

....unique closure operators can be defined on any n element set, n 10. They have been studied in chordal and block graphs [17,10] and in partial orders [22] Antimatroid closure spaces are important because they support greedy algorithms, many of which are examined as shelling operators in [19] In [23], it was shown that the subsets of any closure space (U; could be ordered by X Y iff Y X: X Y: 1) The resulting partial order on all the subsets of U is a lattice L on the 2 jU j elements of the power set, as shown in Figure 1(b) provided is antimatroid. Here the sublattice of ....

....discovered that the sublattice C of closed sets is semi modular [20] and if every singleton is closed it is atomic as well. The entire lattice L is neither semi modular nor complemented nor atomic; but it is very regular . Two properties we will use in this paper were demonstrated in [23]. Theorem 1 (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) ....

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John L. Pfaltz. Closure Lattices. Discrete Mathematics, 154:217--236, 1996.

....phrase structure grammars in that intermediate symbols are not rewritten. Instead, completely new pieces are simply added to the discrete system being generated. Such discrete systems we model by graphs, G k , in this paper. The second theme is that of closure spaces and their associated lattices [19]. If the class of discrete system being generated is a closure space, we will be able to define good grammars as those which homomorphically preserve the induced lattice structure of the system with each step of the generation. Finally, we will be able to develop a hierarchy of grammars that ....

....paths between distinct points p; q 2 X is uniquely generated over chordal graphs. A vector space, or matroid M, is the closure (or span oe) of a set of basis vectors [21] M must satisfy the exchange axiom: p; q 62 X:oe, and q 2 (X [ p) oe imply that 7 p 2 (X [ q) oe. 3 It can be shown [19] that uniquely generated closure spaces must satisfy the anti exchange axiom: p; q 62 X: and q 2 (X [ p) imply that p 62 (X [ q) From this comes the adjective antimatroid closure space. Closure spaces form a kind of dual to vector spaces. If the sets of a closure space (U; are partially ....

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John L. Pfaltz. Closure Lattices. Discrete Mathematics, 154:217--236, 1996.

....= X: X: Y: implies (X Y ) X: Y: The last axiom is non standard. It is not hard to show that closure operators which satisfy this additional axiom are uniquely generated in the sense that for any set Y , there exists a unique minimal set X Y such that X: Y: One can also show [16] that Theorem 1.1 A closure operator is uniquely generated if and only if it satisfies the antiexchange property if p; q 62 X: then p 2 (X [ fqg) implies q 62 (X [ fpg) In contrast, any set of elements U with an operator oe satisfying the first three closure axioms, together with the ....

.... closure axioms, together with the Steinitz MacLane exchange axiom if p; q 62 X:oe then p 2 (X [ fqg) oe implies q 2 (X [ fpg) oe is called a matroid [10] 17] 2] 1 Any set U and closure operator satisfying an antiexchange axiom, U; is called an antimatroid [3] 9] or closure space [16]. 2 Other common names for this concept are APS greedoid, shelling structure [7] alignment [6] or convex geometry [5] provided only that one further requires the empty set, to be closed. 1 The closure operator oe of a matroid is normally called the spanning operator. 2 Closure spaces ....

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John L. Pfaltz. Closure Lattices. Discrete Mathematics, 154:217--236, 1996.

....= Z is said to be closed. A minimal subset X Z such that X: Z: is called a generator, or a kernel 2 and denoted Z: The anti matroid properties come from a fourth anti exchange axiom, p; q 62 Y: and q 2 (Y [ fpg) imply p 62 (Y [ fqg) Many of these properties have been described in [14]. In particular it is shown that in an anti matroid closure space every set is uniquely generated. If instead the closure system satisfies the exchange axiom p; q 62 Y: and q 2 (Y [ fpg) imply p 2 (Y [ fqg) one has a matroid, or generalized vector space. Matroid properties have been ....

....the literature for these minimal generating sets depending on ones approach. With convex closure in discrete geometry one speaks of extreme points [4] a term we will use in this paper as well. With respect to transitive closures in relational algebras one calls them the irreducible kernel [2] In [14], we called them generators, but denoted them with the symbol fi, suggestive of basis the generators of a matroid space. 3 One must be careful. Shortest paths in chordal graphs are not unique closure generators; but monophonic paths are [5] 2 are finitely generated. However, there are two ....

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John L. Pfaltz. Closure Lattices. Discrete Mathematics, 154:217--236, 1996.

....space to mean an antimatroid closure space. The last axiom is non standard. It is not hard to show that closure operators which satisfy this additional axiom are uniquely generated in the sense that for any set Y , there exists a unique minimal set X Y such that X: Y: One can also show [14] that Theorem 1.1 A closure operator is uniquely generated if and only if it satisfies the antiexchange property if p; q 62 Y: then p 2 (Y [ fqg) implies q 62 (Y [ fpg) In contrast, any set of elements U with an operator oe satisfying the first three closure axioms, together with the ....

....spaces provided n 10 [12] Similarly, there are many different closure operators, 3 The term basis has so many connotations, especially with respect to vector spaces and their change of basis, that we prefer the more neutral generator . 2 Antimatroid closure spaces have been studied in [14], in which the subsets X;Y U are partially ordered by , where X Y if and only if Y X: X Y: 1) This is a partial order on all the subsets of U, not just its closed subsets. It is possible to show that this partial ordering of 2 U is, in fact, a well structured lattice, L, called ....

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John L. Pfaltz. Closure Lattices. Discrete Mathematics, 154:217--236, 1996.

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