Antimatroid

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Three views of an antimatroid: an inclusion ordering on its family of feasible sets, a formal language, and the corresponding path poset.

In mathematics, an antimatroid is a formal system that describes processes in which a set is built up by including elements one at a time, and in which an element, once available for inclusion, remains available until it is included. Antimatroids are commonly axiomatized in two equivalent ways, either as a set system modeling the possible states of such a process, or as a formal language modeling the different sequences in which elements may be included. Dilworth (1940) was the first to study antimatroids, taking a lattice theoretic point of view, and they have been frequently rediscovered in other contexts;^[1] see Korte et al. (1991) for a comprehensive survey of antimatroid theory with many additional references.

The axioms defining antimatroids as set systems have some similarity to those of matroids, but whereas matroids are defined by an exchange axiom, antimatroids can be defined instead by an anti-exchange axiom, from which their name derives. Antimatroids can be viewed as a special case of either greedoids or semimodular lattices; one characterization of antimatroids is that they are greedoids such that the inclusion ordering on their feasible sets forms a semimodular lattice.^[2] Antimatroids generalize partial orders and distributive lattices. Antimatroids are also complementary to convex geometries, a combinatorial abstraction of convex sets in geometry.

Glasserman and Yao (1994) use antimatroids to model the ordering of events in discrete event simulation systems, and Parmar (2003) uses them to model progress towards a goal in artificial intelligence planning problems. In mathematical psychology, antimatroids have been used to describe feasible states of knowledge of a human learner.^[3]

The number of possible antimatroids on a set of elements grows rapidly with the number of elements in the set. For sets of one, two, three, etc. elements, the number of possible antimatroids is

1, 3, 22, 485, 59386, ... (sequence A119770 in OEIS).

[edit] Definitions

An antimatroid can be defined as a finite family F of sets, called feasible sets, with the following two properties:

The union of any two feasible sets is also feasible. That is, F is closed under unions.
If S is a nonempty feasible set, then there exists some x in S such that S \ {x} (the set formed by removing x from S) is also feasible. That is, F is an accessible set system.

Antimatroids also have an equivalent definition as a formal language, that is, as a set of strings defined from a finite alphabet of symbols. A language L defining an antimatroid must satisfy the following properties:

Every symbol of the alphabet occurs in at least one word of L. That is, in the terminology of Korte et al., L is normal.
No string in L contains two copies of the same symbol. That is, in the terminology of Korte et al., L is simple.
Every prefix of a string in L is also in L. That is, in the terminology of Korte et al., L is hereditary.
If s and t are strings in L, and the set of symbols in s is not a subset of the set of symbols in t, then there is a symbol x in s such that tx is another string in L.

If L is an antimatroid defined as a formal language, then one can derive from it an accessible union-closed set system F, where F consists of the sets of symbols in strings of L. In the other direction, if F is an accessible union-closed set system, and L is the language of strings s with the property that the set of symbols in each prefix of s is feasible, then L defines an antimatroid. Thus, these two definitions lead to mathematically equivalent classes of objects.^[4]

[edit] Examples

A shelling sequence of a planar point set. The line segments show edges of the convex hulls after some of the points have been removed.

A chain antimatroid is defined by a total order on its elements; the nonempty sets in the antimatroid are all the sets of the form {y | y ≤ x} for some element x. Thus, a chain antimatroid has exactly one feasible set of each different possible cardinality. A chain antimatroid can also be defined as a formal language, as the set of prefixes of a single repetition-free word.

A poset antimatroid has as its feasible sets the lower sets of a finite partially ordered set. A chain antimatroid is a special case of a poset antimatroid. By Birkhoff's representation theorem for distributive lattices, the feasible sets in a poset antimatroid (ordered by set inclusion) form a distributive lattice, and any distributive lattice can be formed in this way. Thus, antimatroids can be seen as generalizations of distributive lattices. Any antimatroid is the homomorphic image of the poset antimatroid of its path poset.^[5]

A different antimatroid that may be constructed from any finite partial order has as its feasible sets the unions of upper sets and lower sets. The complements of these sets are the convex subsets of the partial order.^[6]

A shelling sequence of a finite set U of points in the Euclidean plane or a higher dimensional Euclidean space is an ordering on the points such that, for each point p, there is a line (in the Euclidean plane, or a hyperplane in a Euclidean space) that separates p from all later points in the sequence. Equivalently, p must be a vertex of the convex hull of it and all later points. The partial shelling sequences of a point set form an antimatroid, called a shelling antimatroid. The feasible sets of the shelling antimatroid are the intersections of U with the complement of a convex set. As can be seen in the figure, the edges of the convex hulls formed by this shelling process together form a graph that is a pointed pseudotriangulation.

A perfect elimination ordering of a chordal graph is an ordering of its vertices such that, for each vertex v, the neighbors of v that occur later than v in the ordering form a clique. The prefixes of perfect elimination orderings of a chordal graph form an antimatroid.^[7]

[edit] Paths and basic words

In the set theoretic axiomatization of an antimatroid we may identify a subfamily of special sets, called paths, that determine the whole antimatroid: every set in the antimatroid is a union of paths. An element x that can be removed from a feasible set S to form another feasible set is called an endpoint of S, and a feasible set that has only one endpoint is called a path of the antimatroid. The subset ordering of the paths forms a partially ordered set, called the path poset of the antimatroid.

For every feasible set S in the antimatroid, and every element x of S, one may find a path subset of S for which x is an endpoint: to do so, remove one at a time elements other than x until no such removal leaves a feasible subset. Therefore, each feasible set in an antimatroid is the union of its path subsets. If S is not a path, each subset in this union is a proper subset of S. But, if S is itself a path with endpoint x, each proper subset of S that belongs to the antimatroid excludes x. Therefore, the paths of an antimatroid are exactly the sets that do not equal the unions of their proper subsets in the antimatroid. Equivalently, a given family of sets P forms the set of paths of an antimatroid if and only if, for each S in P, the union of subsets of S in P has one fewer element than S itself. If so, F itself is the family of unions of subsets of P.

In the formal language formalization of an antimatroid we may also identify a subset of words that determine the whole language, the basic words. The longest strings in L are called basic words; each basic word forms a permutation of the whole alphabet. For instance, the basic words of a poset antimatroid are the linear extensions of the given partial order. If B is the set of basic words, L can be defined from B as the set of prefixes of words in B. It is often convenient to define antimatroids from basic words in this way, but it is not straightforward to write an axiomatic definition antimatroids in terms of their basic words.

[edit] Convex geometries

If F is the set system defining an antimatroid, with U equal to the union of the sets in F, then the family of sets

$G = \{U\setminus S\mid S\in F\}$

complementary to the sets in F is sometimes called a convex geometry, and the sets in G are called convex sets. For instance, in a shelling antimatroid, the convex sets are intersections of U with convex subsets of the Euclidean space into which U is embedded.

Complementarily to the properties of set systems that define antimatroids, the set system defining a convex geometry should be closed under intersections, and for any set S in G that is not equal to U there must be an element x not in S that can be added to S to form another set in G.

A convex geometry can also be defined in terms of a closure operator τ that maps any subset of U to its minimal closed superset. To be a closure operator, τ should have the following properties:

τ(Ø) = Ø: the closure of the empty set is empty.
Any set S is a subset of τ(S).
If S is a subset of T, then τ(S) must be a subset of τ(T).
For any set S, τ(S) = τ(τ(S)).

The family of closed sets resulting from a closure operation of this type is necessarily closed under intersections. The closure operators that define convex geometries also satisfy an additional anti-exchange axiom:

If neither y nor z belong to τ(S), but z belongs to τ(S ∪ {y}), then y does not belong to τ(S ∪ {z}).

A closure operation satisfying this axiom is called an anti-exchange closure. If S is a closed set in an anti-exchange closure, then the anti-exchange axiom determines a partial order on the elements not belonging to S, where x ≤ y in the partial order when x belongs to τ(S ∪ {y}). If x is a minimal element of this partial order, then S ∪ {x} is closed. That is, the family of closed sets of an anti-exchange closure has the property that for any set other than the universal set there is an element x that can be added to it to produce another closed set. This property is complementary to the accessibility property of antimatroids, and the fact that intersections of closed sets are closed is complementary to the property that unions of feasible sets in an antimatroid are feasible. Therefore, the complements of the closed sets of any anti-exchange closure form an antimatroid.^[8]

[edit] Join-distributive lattices

Any two sets in an antimatroid have a unique least upper bound (their union) and a unique greatest lower bound (the union of the sets in the antimatroid that are contained in both of them). Therefore, the sets of an antimatroid, partially ordered by set inclusion, form a lattice. Various important features of an antimatroid can be interpreted in lattice-theoretic terms; for instance the paths of an antimatroid are the join-irreducible elements of the corresponding lattice, and the basic words of the antimatroid correspond to maximal chains in the lattice. The lattices that arise from antimatroids in this way generalize the finite distributive lattices, and can be characterized in several different ways.

The description originally considered by Dilworth (1940) concerns meet-irreducible elements of the lattice. For each element x of an antimatroid, there exists a unique maximal feasible set S_x that does not contain x (S_x is the union of all feasible sets not containing x). S_x is meet-irreducible, meaning that it is not the meet of any two larger lattice elements: any larger feasible set, and any intersection of larger feasible sets, contains x and so does not equal S_x. Any element of any lattice can be decomposed as a meet of meet-irreducible sets, often in multiple ways, but in the lattice corresponding to an antimatroid each element T has a unique minimal family of meet-irreducible sets S_x whose meet is T; this family consists of the sets S_x such that T ∪ {x} belongs to the antimatroid. That is, the lattice has unique meet-irreducible decompositions.

A second characterization concerns the intervals in the lattice, the sublattices defined by a pair of lattice elements x ≤ y and consisting of all lattice elements z with x ≤ z ≤ y. An interval is atomistic if every element in it is the join of atoms (the minimal elements above the bottom element x), and it is boolean if it is isomorphic to the lattice of all subsets of a finite set. For an antimatroid, every interval that is atomistic is also boolean.

Thirdly, the lattices arising from antimatroids are semimodular lattices, lattices that satisfy the upper semimodular law that for any two elements x and y, if y covers x ∧ y then x ∨ y covers x. Translating this condition into the sets of an antimatroid, if a set Y has only one element not belonging to X then that one element may be added to X to form another set in the antimatroid. Additionally, the lattice of an antimatroid has the meet-semidistributive property: for all lattice elements x, y, and z, if x ∧ y and x ∧ z are both equal then they also equal x ∧ (y ∨ z). A semimodular and meet-semidistributive lattice is called a join-distributive lattice.

These three characterizations are equivalent: any lattice with unique meet-irreducible decompositions has boolean atomistic intervals and is join-distributive, any lattice with boolean atomistic intervals has unique meet-irreducible decompositions and is join-distributive, and any join-distributive lattice has unique meet-irreducible decompositions and boolean atomistic intervals.^[9] Thus, we may refer to a lattice with any of these three properties as join-distributive. Any antimatroid gives rise to a finite join-distributive lattice, and any finite join-distributive lattice comes from an antimatroid in this way.^[10] Another equivalent characterization of finite join-distributive lattices is that they are graded (any two maximal chains have the same length), and the length of a maximal chain equals the number of meet-irreducible elements of the lattice.^[11] The antimatroid representing a finite join-distributive lattice can be recovered from the lattice: the elements of the antimatroid can be taken to be the meet-irreducible elements of the lattice, and the feasible set corresponding to any element x of the lattice consists of the set of meet-irreducible elements y such that y is not greater than or equal to x in the lattice.

This representation of any finite join-distributive lattice as an accessible family of sets closed under unions (that is, as an antimatroid) may be viewed as an analogue of Birkhoff's representation theorem under which any finite distributive lattice has a representation as a family of sets closed under unions and intersections.

[edit] Supersolvable antimatroids

Motivated by a problem of defining partial orders on the elements of a Coxeter group, Armstrong (2007) defines a subclass of antimatroids which he calls the supersolvable antimatroids. A supersolvable antimatroid is defined by a totally ordered collection of elements, and a family of sets of these elements. The family must include the empty set. Additionally, it must have the property that if two sets A and B belong to the family, the set-theoretic difference B \ A is nonempty, and x is the smallest element of B \ A, then A ∪ {x} also belongs to the family. As Armstrong observes, any family of sets of this type forms an antimatroid. Armstrong also provides a lattice-theoretic characterization of the antimatroids that this construction can form.

[edit] Join operation and convex dimension

If A and B are two antimatroids, both described as a family of sets, and if the maximal sets in A and B are equal, we can form another antimatroid, the join of A and B, as follows:

$A\vee B = \{ S\cup T \mid S\in A \wedge T\in B \}.$

Note that this is a different operation than the join considered in the lattice-theoretic characterizations of antimatroids: it combines two antimatroids to form another antimatroid, rather than combining two sets in an antimatroid to form another set. The family of all antimatroids that have a given maximal set forms a semilattice with this join operation.

Joins are closely related to a closure operation that maps formal languages to antimatroids, where the closure of a language L is the intersection of all antimatroids containing L as a sublanguage. This closure has as its feasible sets the unions of prefixes of strings in L. In terms of this closure operation, the join is the closure of the union of the languages of A and B.

Every antimatroid can be represented as a join of a family of chain antimatroids, or equivalently as the closure of a set of basic words; the convex dimension of an antimatroid A is the minimum number of chain antimatroids (or equivalently the minimum number of basic words) in such a representation. If F is a family of chain antimatroids whose basic words all belong to A, then F generates A if and only if the feasible sets of F include all paths of A. The paths of A belonging to a single chain antimatroid must form a chain in the path poset of A, so the convex dimension of an antimatroid equals the minimum number of chains needed to cover the path poset, which by Dilworth's theorem equals the width of the path poset.^[12]

If one has a representation of an antimatroid as the closure of a set of d basic words, then this representation can be used to map the feasible sets of the antimatroid into d-dimensional Euclidean space: assign one coordinate per basic word w, and make the coordinate value of a feasible set S be the length of the longest prefix of w that is a subset of S. With this embedding, S is a subset of T if and only if the coordinates for S are all less than or equal to the corresponding coordinates of T. Therefore, the order dimension of the inclusion ordering of the feasible sets is at most equal to the convex dimension of the antimatroid.^[13] However, in general these two dimensions may be very different: there exist antimatroids with order dimension three but with arbitrarily large convex dimension.

[edit] Notes

^ Monjardet (1985).
^ E.g., see Gordon (1997).
^ See e.g. Doignon & Falmagne (1999), who refer to structures equivalent to antimatroids as "well-graded knowledge spaces".
^ Korte et al., Theorem 1.4.
^ Korte et al., Corollary 6.2.
^ Birkhoff & Bennett (1985).
^ Gordon (1997) describes several results related to antimatroids of this type, but these antimatroids were mentioned earlier e.g. by Korte et al. Chandran et al. (2003) use the connection to antimatroids as part of an algorithm for efficiently listing all perfect elimination orderings of a given chordal graph.
^ Korte et al., Theorem 1.1.
^ Adaricheva, Gorbunov & Tumanov (2003), Theorems 1.7 and 1.9; Armstrong (2007), Theorem 2.7.
^ Edelman (1980), Theorem 3.3; Armstrong (2007), Theorem 2.8.
^ Monjardet (1985) credits a dual form of this characterization to several papers from the 1960s by S. P. Avann.
^ Edelman & Saks (1988); Korte et al., Theorem 6.9.
^ Korte et al., Corollary 6.10.

[edit] References

Adaricheva, K. V.; Gorbunov, V. A.; Tumanov, V. I. (2003), "Join-semidistributive lattices and convex geometries", Advances in Mathematics 173 (1): 1–49, doi:10.1016/S0001-8708(02)00011-7 .

Armstrong, Drew (2007), The sorting order on a Coxeter group, arΧiv:0712.1047 .

Birkhoff, Garret; Bennett, M. K. (1985), "The convexity lattice of a poset", Order 2 (3): 223–242, doi:10.1007/BF00333128 .

Chandran, L. S.; Ibarra, L.; Ruskey, F.; Sawada, J. (2003), "Generating and characterizing the perfect elimination orderings of a chordal graph", Theoretical Computer Science 307: 303–317, doi:10.1016/S0304-3975(03)00221-4, http://old-www.cis.uoguelph.ca/~sawada/papers/chordal.pdf .

Dilworth, Robert P. (1940), "Lattices with unique irreducible decompositions", Annals of Mathematics 41: 771–777, doi:10.2307/1968857 .

Doignon, Jean-Paul; Falmagne, Jean-Claude (1999), Knowledge Spaces, Springer-Verlag, ISBN 3-540-64501-2

Edelman, Paul H. (1980), "Meet-distributive lattices and the anti-exchange closure", Algebra Universalis 10 (1): 290–299, doi:10.1007/BF02482912 .

Edelman, Paul H.; Saks, Michael E. (1988), "Combinatorial representation and convex dimension of convex geometries", Order 5 (1): 23–32, doi:10.1007/BF00143895 .

Glasserman, Paul; Yao, David D. (1994), Monotone Structure in Discrete Event Systems, Wiley Series in Probability and Statistics, Wiley Interscience, ISBN 978-0471580416 .

Gordon, Gary (1997), "A β invariant for greedoids and antimatroids", Electronic Journal of Combinatorics 4 (1): Research Paper 13, MR 1445628, http://www.combinatorics.org/Volume_4/Abstracts/v4i1r13.html .

Korte, Bernard; Lovász, László; Schrader, Rainer (1991), Greedoids, Springer-Verlag, pp. 19–43, ISBN 3-540-18190-3 .

Monjardet, Bernard (1985), "A use for frequently rediscovering a concept", Order 1 (4): 415–417, doi:10.1007/BF00582748 .

Parmar, Aarati (2003), "Some Mathematical Structures Underlying Efficient Planning", AAAI Spring Symposium on Logical Formalization of Commonsense Reasoning, http://www-formal.stanford.edu/aarati/papers/SS603AParmar.pdf .

[0] Monjardet (1985).

[1] E.g., see Gordon (1997).

[2] See e.g. Doignon & Falmagne (1999), who refer to structures equivalent to antimatroids as "well-graded knowledge spaces".

[3] Korte et al., Theorem 1.4.

[4] Korte et al., Corollary 6.2.

[5] Birkhoff & Bennett (1985).

[6] Gordon (1997) describes several results related to antimatroids of this type, but these antimatroids were mentioned earlier e.g. by Korte et al. Chandran et al. (2003) use the connection to antimatroids as part of an algorithm for efficiently listing all perfect elimination orderings of a given chordal graph.

[7] Korte et al., Theorem 1.1.

[8] Adaricheva, Gorbunov & Tumanov (2003), Theorems 1.7 and 1.9; Armstrong (2007), Theorem 2.7.

[9] Edelman (1980), Theorem 3.3; Armstrong (2007), Theorem 2.8.

[10] Monjardet (1985) credits a dual form of this characterization to several papers from the 1960s by S. P. Avann.

[11] Edelman & Saks (1988); Korte et al., Theorem 6.9.

[12] Korte et al., Corollary 6.10.

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Antimatroid

From Wikipedia, the free encyclopedia

Contents

[edit] Definitions

[edit] Examples

[edit] Paths and basic words

[edit] Convex geometries

[edit] Join-distributive lattices

[edit] Supersolvable antimatroids

[edit] Join operation and convex dimension

[edit] Notes

[edit] References

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