Antimatroid

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, using yet another axiomatization based on lattice theory, and they have been frequently rediscovered in other contexts; see for a comprehensive survey of antimatroid theory with many additional references.
The axioms defining antimatroids as set systems are very similar to those of matroids, but whereas matroids are defined by an "exchange axiom" (e.g., the "basis exchange", or "independent set exchange" axioms), antimatroids are defined instead by an "anti-exchange axiom", from which their name derives.
Antimatroids can be viewed as a special case of greedoids and of semimodular lattices, and as a generalization of partial orders and of distributive lattices. 
Antimatroids are equivalent, by complementation, to convex geometries, a combinatorial abstraction of convex sets in geometry.
Antimatroids have been applied to model precedence constraints in scheduling problems, potential event sequences in simulations, task planning in artificial intelligence, and the states of knowledge of human learners.
Definitions.
An antimatroid can be defined as a finite family formula_1 of sets, called "feasible sets", with the following two properties:
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 string that belongs to this set is called a "word" of the language. A language formula_10 defining an antimatroid must satisfy the following properties:
The equivalence of these two forms of definition can be seen as follows. If formula_10 is an antimatroid defined as a formal language, then the sets of symbols in words of formula_10 form an accessible union-closed set system. It is accessible by the hereditary property of strings, and it can be shown to be union-closed by repeated application of the concatenation property of strings. In the other direction, if formula_1 is an accessible union-closed set system, one can interpret the elements of its sets as symbols in an alphabet. For this alphabet, one can define a language formula_10 consisting of all strings formula_3 that have at most one copy of each symbol and that obey the have the property that every prefix of formula_3 has a set of symbols that belongs to formula_1. This language automatically obeys all of the properties of a formal language that are required for an antimatroid. These two transformations are the inverses of each other: transforming a formal language into a set family and back, or vice versa, produces the same system. Thus, these two definitions lead to mathematically equivalent classes of objects.
Paths and basic words.
In the set theoretic axiomatization of an antimatroid there are certain special sets called "paths" that determine the whole antimatroid, in the sense that the sets of the antimatroid are exactly the unions of paths. If formula_3 is any feasible set of the antimatroid, an element formula_5 that can be removed from formula_3 to form another feasible set is called an "endpoint" of formula_3, and a feasible set that has only one endpoint is called a "path" of the antimatroid. The family of paths can be partially ordered by set inclusion, forming the "path poset" of the antimatroid.
For every feasible set formula_3 in the antimatroid, and every element formula_5 of formula_3, one may find a path subset of formula_3 for which formula_5 is an endpoint: to do so, remove one at a time elements other than formula_5 until no such removal leaves a feasible subset. Therefore, each feasible set in an antimatroid is the union of its path subsets. If formula_3 is not a path, each subset in this union is a proper subset of formula_3. But, if formula_3 is itself a path with endpoint formula_5, each proper subset of formula_3 that belongs to the antimatroid excludes formula_5. Therefore, the paths of an antimatroid are exactly the feasible sets that do not equal the unions of their proper feasible subsets. Equivalently, a given family of sets formula_59 forms the family of paths of an antimatroid if and only if, for each formula_3 in formula_59, the union of subsets of formula_3 in formula_59 has one fewer element than formula_3 itself. If so, formula_1 itself is the family of unions of subsets of formula_59.
In the formal language formalization of an antimatroid we may also identify a subset of words that determine the whole language, the "basic words".
If formula_10 is the formal language of an antimatroid, the longest strings in formula_10 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 formula_69 is the set of basic words, formula_10 can be defined from formula_69 as the set of prefixes of words in formula_69.
Convex geometries.
If formula_1 is the set system defining an antimatroid, with formula_35 equal to the union of the sets in formula_1, then the family of sets
formula_76
complementary to the sets in formula_1 is sometimes called a convex geometry and the sets in formula_78 are called convex sets. For instance, in a shelling antimatroid, the convex sets are intersections of formula_35 with convex subsets of the Euclidean space into which formula_35 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 formula_3 in formula_78 that is not equal to formula_35 there must be an element formula_5 not in formula_3 that can be added to formula_3 to form another set in formula_78.
A convex geometry can also be defined in terms of a closure operator formula_88 that maps any subset of formula_35 to its minimal closed superset. To be a closure operator, formula_88 should have the following properties:
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:
A closure operation satisfying this axiom is called an anti-exchange closure. If formula_3 is a closed set in an anti-exchange closure, then the anti-exchange axiom determines a partial order on the elements not belonging to formula_3, where formula_112 in the partial order when formula_5 belongs to formula_107. If formula_5 is a minimal element of this partial order, then formula_116 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 formula_5 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.
The undirected graphs in which the convex sets (subsets of vertices that contain all shortest paths between vertices in the subset) form a convex geometry are exactly the Ptolemaic graphs.
Join-distributive lattices.
Every two feasible sets of 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 feasible 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.
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. 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. 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. 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 formula_5 of the lattice consists of the set of meet-irreducible elements formula_102 such that formula_102 is not greater than or equal to formula_5 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.
Supersolvable antimatroids.
Motivated by a problem of defining partial orders on the elements of a Coxeter group, studied antimatroids which are also supersolvable lattices. 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 formula_154 and formula_69 belong to the family, the set-theoretic difference formula_156 is nonempty, and formula_5 is the smallest element of formula_156, then formula_159 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.
Join operation and convex dimension.
If formula_160 and formula_161 are two antimatroids, both described as a family of sets over the same universe of elements, we can form another antimatroid, the "join" of formula_160 and formula_161, as follows:
formula_164
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 over the same universe 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 formula_10 is the intersection of all antimatroids containing formula_10 as a sublanguage. This closure has as its feasible sets the unions of prefixes of strings in formula_10. In terms of this closure operation, the join is the closure of the union of the languages of formula_160 and formula_161. 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 formula_160 is the minimum number of chain antimatroids (or equivalently the minimum number of basic words) in such a representation. If formula_171 is a family of chain antimatroids whose basic words all belong to formula_160, then formula_171 generates formula_160 if and only if the feasible sets of formula_171 include all paths of formula_160. The paths of formula_160 belonging to a single chain antimatroid must form a chain in the path poset of formula_160, 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.
If one has a representation of an antimatroid as the closure of a set of formula_179 basic words, then this representation can be used to map the feasible sets of the antimatroid to points in formula_179-dimensional Euclidean space: assign one coordinate per basic word formula_181, and make the coordinate value of a feasible set formula_3 be the length of the longest prefix of formula_181 that is a subset of formula_3. With this embedding, formula_3 is a subset of another feasible set formula_16 if and only if the coordinates for formula_3 are all less than or equal to the corresponding coordinates of formula_16. Therefore, the order dimension of the inclusion ordering of the feasible sets is at most equal to the convex dimension of the antimatroid. However, in general these two dimensions may be very different: there exist antimatroids with order dimension three but with arbitrarily large convex dimension.
Enumeration.
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 distinct antimatroids is
formula_189
Applications.
Both the precedence and release time constraints in the standard notation for theoretic scheduling problems may be modeled by antimatroids. use antimatroids to generalize a greedy algorithm of Eugene Lawler for optimally solving single-processor scheduling problems with precedence constraints in which the goal is to minimize the maximum penalty incurred by the late scheduling of a task.
 use antimatroids to model the ordering of events in discrete event simulation systems.
 uses antimatroids to model progress towards a goal in artificial intelligence planning problems.
In Optimality Theory, a mathematical model for the development of natural language based on optimization under constraints, grammars are logically equivalent to antimatroids.
In mathematical psychology, antimatroids have been used to describe feasible states of knowledge of a human learner. Each element of the antimatroid represents a concept that is to be understood by the learner, or a class of problems that he or she might be able to solve correctly, and the sets of elements that form the antimatroid represent possible sets of concepts that could be understood by a single person. The axioms defining an antimatroid may be phrased informally as stating that learning one concept can never prevent the learner from learning another concept, and that any feasible state of knowledge can be reached by learning a single concept at a time. The task of a knowledge assessment system is to infer the set of concepts known by a given learner by analyzing his or her responses to a small and well-chosen set of problems. In this context antimatroids have also been called "learning spaces" and "well-graded knowledge spaces".