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 was the first to study antimatroids, using yet another axiomatization based on lattice theory, and they have been frequently rediscovered in other contexts.
The axioms defining antimatroids as set systems are very similar to those of matroids, but whereas matroids are defined by an exchange axiom, 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 of finite sets, called feasible sets, with the following two properties:

The union of any two feasible sets is also feasible. That is, is closed under unions.
If is a nonempty feasible set, then contains an element for which is also feasible. That is, 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 string that belongs to this set is called a word of the language. A language defining an antimatroid must satisfy the following properties:

Every symbol of the alphabet occurs in at least one word of.
Each word of contains at most one copy of each symbol. A language with this property is called normal.
Every prefix of a word in is also in. A language with this property is called hereditary.
If and are words in, and contains at least one symbol that is not in, then there is a symbol in such that the concatenation is another word in.

The equivalence of these two forms of definition can be seen as follows. If is an antimatroid defined as a formal language, then the sets of symbols in words of 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, from an accessible union-closed set system , the language of normal strings whose prefixes all have sets of symbols belonging to meets the requirements for a formal language to be 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.

Examples

The following systems provide examples of antimatroids:
;Chain antimatroids
;Poset antimatroids
;Shelling antimatroids
;Perfect elimination
;Chip-firing games

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 is any feasible set of the antimatroid, an element that can be removed from to form another feasible set is called an endpoint of, 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 in the antimatroid, and every element of, one may find a path subset of for which is an endpoint: to do so, remove one at a time elements other than until no such removal leaves a feasible subset. Therefore, each feasible set in an antimatroid is the union of its path subsets. If is not a path, each subset in this union is a proper subset of. But, if is itself a path with endpoint, each proper subset of that belongs to the antimatroid excludes. 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 forms the family of paths of an antimatroid if and only if, for each in, the union of subsets of in has one fewer element than itself. If so, itself is the family of unions of subsets of.
In the formal language formalization of an antimatroid, the longest strings are called basic words. Each basic word forms a permutation of the whole alphabet. If is the set of basic words, can be defined from as the set of prefixes of words in.

Convex geometries

If is the set system defining an antimatroid, with equal to the union of the sets in, then the family of sets
complementary to the sets in is sometimes called a convex geometry and the sets in are called convex sets. For instance, in a shelling antimatroid, the convex sets are intersections of the given point set with convex subsets of Euclidean space. The set system defining a convex geometry must be closed under intersections. For any set in that is not equal to there must be an element not in that can be added to to form another set in.
A convex geometry can also be defined in terms of a closure operator that maps any subset of to its minimal closed superset. To be a closure operator, should have the following properties:

: the closure of the empty set is empty.
For every subset of, is a subset of and.
Whenever, is a subset of.

The family of closed sets resulting from a closure operation of this type is necessarily closed under intersections, but might not be a convex geometry. The closure operators that define convex geometries also satisfy an additional anti-exchange axiom:

If is a subset of, and and are distinct elements of that do not belong to, but does belong to, then does not belong to.

A closure operation satisfying this axiom is called an anti-exchange closure. If is a closed set in an anti-exchange closure, then the anti-exchange axiom determines a partial order on the elements not belonging to, where in the partial order when belongs to. If is a minimal element of this partial order, then 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 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 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 and a unique greatest lower bound. 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.

The description originally considered by concerns meet-irreducible elements of the lattice. For each element of an antimatroid, there exists a unique maximal feasible set that does not contain : can be constructed as the union of all feasible sets not containing. This set is automatically meet-irreducible, meaning that it is not the meet of any two larger lattice elements. This is true because every feasible superset of contains, and the same is therefore also true of every intersection of feasible supersets. Every element of an arbitrary 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 has a unique minimal family of meet-irreducible sets whose meet is ; this family consists of the sets for the elements such that is feasible. 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 consisting of all lattice elements with. An interval is atomistic if every element in it is the join of atoms, 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 every two elements and, if covers then covers. Translating this condition into the feasible sets of an antimatroid, if a feasible set has only one element not belonging to another feasible set then that one element may be added to to form another set in the antimatroid. Additionally, the lattice of an antimatroid has the meet-semidistributive property: for all lattice elements,, and, if and equal each other then they also both equal. 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. 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, 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 of the lattice consists of the set of meet-irreducible elements such that is not greater than or equal to in the lattice.
This representation of any finite join-distributive lattice as an accessible family of sets closed under unions 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.