Poly-Dimension of Antimatroids

A partial cube is a graph that can be isometrically embedded into a hypercube. In other words, a partial cube is a subgraph of a hypercube that preserves distances the distance between any two vertices in the subgraph is the same as the distance between those vertices in the hypercube. Partial cubes were first introduced by Graham and Pollak (Graham & Pollak, 1971) as a model for communication networks and were extensively studied afterwards.


Introduction
A partial cube is a graph that can be isometrically embedded into a hypercube. In other words, a partial cube is a subgraph of a hypercube that preserves distances -the distance between any two vertices in the subgraph is the same as the distance between those vertices in the hypercube. Partial cubes were first introduced by Graham and Pollak (Graham & Pollak, 1971) as a model for communication networks and were extensively studied afterwards.
Many important families of combinatorial structures can be represented as subgraphs of partial cubes, for instance, antimatroids. An antimatroid is an accessible set system closed under union. There are two equivalent definitions of antimatroids, one as set systems and the other as languages (Björner & Ziegler, 1992;Korte et al., 1991). An algorithmic characterization of antimatroids based on the language definition was introduced in (Boyd & Faigle, 1990). Later, another algorithmic characterization of antimatroids, which depicted them as set systems was developed in (Kempner & Levit, 2003). Dilworth (Dilworth, 1940) was the first to study antimatroids using another axiomatization based on lattice theory, and they have been frequently rediscovered in other contexts. An antimatroid can be viewed as a special case of either greedoids or semimodular lattices, and as a generalization of partial orders and distributive lattices. While classical examples of antimatroids connect them with posets, chordal graphs, convex geometries etc., a game theory gives a framework, in which antimatroids are considered as permission structures for coalitions (Algaba et al., 2004;. There are also rich connections between antimatroids and cluster analysis (Kempner & Muchnik, 2003). Glasserman and Yao (Glasserman & Yao, 1994) used antimatroids to model the ordering of events in discrete event simulation systems. In mathematical psychology, antimatroids are used to describe feasible states of knowledge of a human learner (Cosyn & Uzun, 2009;Eppstein et al., 2008;Falmagne & Doignon, 2011).
The notion of "antimatroid with repetition" was conceived by Bjorner, Lovasz and Shor (Björner et al., 1991) as an extension of the notion of antimatroid in the framework of non-simple languages. Further they were investigated by the name of "poly-antimatroids" (Kempner & Levit, 2007;Nakamura, 2005), where the set system approach was used.
An antimatroid with the ground set of size n may be isometrically embedded into the hypercube {0, 1} n . A poly-antimatroid with the same ground set is isometrically embedded into n-dimensional integer lattice Z n . In this research we concentrate on interrelations between antimatroids and poly-antimatroids and prove that these two structures are isometrically isomorphic.
The poly-dimension of an antimatroid is the minimum dimension d such that the antimatroid is isometrically isomorphic to some d-dimensional poly-antimatroid. This definition is a direct analog of the lattice dimension of graphs (Eppstein, 2005). In this paper the exact characterization of antimatroids of poly-dimension 1 and 2 is given and a conjecture concerning antimatroids of any poly-dimension d is suggested.
This chapter is organized as follows.
Section 1 contains an extended introduction to the theory of antimatroids.
Section 2 gives basic information about distances on graphs and isometric isomorphisms. We concentrate on interrelations between antimatroids and poly-antimatroids. In particular, we construct an isometric isomorphism between these structures.
In Section 3 we introduce the poly-dimension of an antimatroid and prove our main theorem characterizing antimatroids of poly-dimension 2. In addition we present a linear labeling algorithm that isometrically embeds such an antimatroid into the integer lattice Z 2 .
Section 4 discusses an open problem.

Preliminaries
Let E be a finite set. A set system over E is a pair (E, S ),w h e r eS is a family of sets over E, called feasible sets. We will use X ∪ x for X ∪{x},andX − x for X −{x}.
Any set system satisfying (A1) is called accessible.
In addition, we use the following characterization of antimatroids.
Proposition 2.2. (Korte et al., 1991) For an accessible set system (E, S ) the following statements are equivalent: An "antimatroid with repetition" was invented by Bjorner, Lovasz and Shor (Björner et al., 1991) by studying the set of configurations of the Chip Firing Game (CFG). Further it was investigated by the name of "poly-antimatroid" as a generalization of the notion of the antimatroid for multisets. A multiset A over E is a function f A : E → N,w h e r e f A (e) is a number of repetitions of an element e in A. A poly-antimatroid is a finite non-empty multiset system (E, F ) that satisfies the antimatroid properties (A1) and (A2). So antimatroids may be considered as a particular case of poly-antimatroids. An example of an antimatroid ({x, y, z}, S ) and a poly-antimatroid ({x, y}, F ) is illustrated in Figure 1. Definition 2.3. A multiset system (E, F ) satisfies the chain property if for all X, Y ∈F,andX⊂ Y, there exists a chain X = X 0 ⊂ X 1 ⊂ ... ⊂ X k = Ys u c ht h a tX i = X i−1 ∪ x i and X i ∈Ffor 0 ≤ i ≤ k. We call the system a chain system.  It is easy to see that this chain property follows from (A2), but these properties are not equivalent. If ∅ ∈F, then accessibility follows from the chain property. In general case, there are accessible set systems that do not satisfy the chain property. Indeed, consider the following example illustrated in Figure 2  In fact, if we have an accessible set system satisfying the chain property, then the same system but without the empty set (or without all subsets of cardinality less then some k)i sn o t accessible, but still satisfies the chain property.
Examples of chain systems include poly-antimatroids, convex geometries, matroids and other hereditary systems (matchings, cliques, independent sets of a graph).
Regular set systems have been defined by Honda and Grabisch (Honda & Grabisch, 2008). Equivalently, under the condition ∅, E ∈S , (E, S ) is a regular set system if and only if |A − B| = 1foranyA, B ∈Ssuch that A is a cover of B.
Actually, a regular set system is a chain system with an additional constraint demanding from both the empty set ∅ and the ground set E to belong to the set system S.
An algorithmic characterization of chain systems with empty set ∅, called strongly accessible set system, was presented in (Boley et al., 2010).
Augmenting systems were introduced in game theory as a restricted cooperation model. The model is a weakening of the antimatroid structure and strengthening of the chain system.
Definition 2.5. (Bilbao, 2003) An augmenting system is an antimatroid if and only if it is closed under union (Bilbao, 2003).
A comparative study of various families of set systems are given in (Grabisch, 2009).
Antimatroids have already been investigated within the framework of lattice theory by Dilworth (Dilworth, 1940). The feasible sets of an antimatroid ordered by inclusion form a lattice, with lattice operations: Since an antimatroid is closed under union, it has only one basis.
A finite lattice L is semimodular if whenever x, y ∈ L both cover x ∧ y, x ∨ y covers x and y.
Proposition 2.6. (Björner & Ziegler, 1992) For an accessible set system (E, S ) the following statements are equivalent: In fact, the two concepts -antimatroids and join-distributive lattices -are essentially equivalent.

) A finite lattice is join-distributive if and only if it is isomorphic to the lattice of feasible sets of some antimatroid.
It is easy to see that feasible sets of a poly-antimatroid ordered by inclusion form a join-distributive lattice as well. This kind of findings was discussed in (Magnien et al., 2001),

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Poly-Dimension of Antimatroids 5 where it was proved that any CFG is equivalent to a simple CFG (antimatroid), i.e., the lattices of their configuration spaces are isomorphic.

Isometry
For each graph G =( V, E) the distance d G (u, v) between two vertices u, v ∈ V is defined as the length of a shortest path joining them.
If G and H are arbitrary graphs, then a mapping f : A family of sets S is well-graded (Doignon & Falmagne, 1997) if any two sets P, Q ∈Scan be connected by a sequence of sets P = R 0 , R 1 , ..., R n = Q formed by single-element insertions and deletions (|R i △ R i+1 | = 1), such that all intermediate sets in the sequence belong to S and |P △ Q| = n.Thissequenceofsetsiscalledatight path.
Any set system (E, S ) defines an undirected graph G S =(S, E S ),whereE S = {{P, Q}∈S: Theorem 3.1. (Ovchinnikov, 2008) The graph G S defined by a set system (E, S ) is a partial cube on E if and only if the family S is well-graded.
Proof. We prove that there is a tight path between each P, Q ∈S.I fP ⊂ Q, then the existence of such path follows immediately from the chain property. If P Q, then there exist chains from both P and Q to P ∪ Q that forms the tight path.
Thus each antimatroid is a partial cube and may be represented as a graph G S that is a subgraph of the hypercube H(E) or as a set of points of the hypercube {0, 1} d . For example, see an antimatroid in Figure 3. The distance in an antimatroid (E, S ) considered as a graph coincides with the Hamming distance between sets, i.e., d S (A, B)=|A △ B| for any A, B ∈S.
A poly-antimatroid (E, F ) may be represented as a set of points in the digital space ) that may be denoted as a point (a 1 , a 2 , ..., a d ) in Z d . For example, see a two-dimensional poly-antimatroid in Figure 4.
The symmetric difference distance can be generalized to multisets by summing the absolute difference of the multiplicities of corresponding elements  that coincides with the L 1 distance (||A − B|| 1 ) on the digital space Z d .
A multiset system (E, F ) defines an undirected graph G F in the same way as a set system. So each poly-antimatroid (E, F ) may be represented as a graph G F .
These representations of poly-antimatroids are isometrically isomorphic. Thus a poly-antimatroid with the ground set of size d may be isometrically embedded into the d-dimensional lattice Z d .
Since each antimatroid is a poly-antimatroid, one may prove that every poly-antimatroid is isometrically isomorphic to some antimatroid.  (Djokovic, 1973) elaborated by Eppstein (Eppstein, 2005) to embed a finite subset of the lattice Z d , and, therefore, a poly-antimatroid (E, F ), into a hypercube.
Note that isometry µ is full-dimensional, i.e., each coordinate µ i takes on both value 0 and 1 for at least one point.

Poly-dimension
Clearly, an antimatroid is a poly-antimatroid as well. The question is what is a minimal dimension d such that an antimatroid (U, S ) may be represented by an isomorphic poly-antimatroid ( U, F ) in the space Z d .
We will call this minimal dimension π (S ) the poly-dimension of the antimatroid (U, S ). Consider antimatroids for which their maximum in-degree and maximum out-degree is at most p, and there is at least one feasible set for which in-degree or out-degree equals p.W e will call such antimatroids p-antimatroids.
To find the poly-dimension of a 2-antimatroid we show that 2-antimatroids have the special structure of "sub-grid".
A poset antimatroid is a particular case of antimatroid, which is formed by the lower sets of a poset (partially ordered set). The poset antimatroids can be characterized as the unique antimatroids which are closed under intersection (Korte et al., 1991).
Let us prove that any 2-antimatroid is a poset antimatroid.

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Poly-Dimension of Antimatroids 9 An endpoint of a feasible set X is an element e ∈ X such that X − e is a feasible set too. A feasible set X is an e-path if it has a single endpoint e.
It is easy to see that an e-path is a minimal feasible set containing e.
The following lemma gives the characterization of poset antimatroids in terms of e-paths. We present a proof for the sake of completeness. If X ∩ Y = ∅ then for every i ∈ U there exists an i-path H i 1 ⊆ X and H i 2 ⊆ Y, because each i-path is a minimal feasible set containing i. By assumption H i   Definition 4.5. A lower zigzag is a sequence of feasible sets P 0 , P 1 , ..., P 2m such that any two consecutive sets in the sequence differ by a single element and P 2i ∈ C k ,a n dP 2i−1 ∈ C k−1 for all 0 ≤ i ≤ m.
In the same way we define an upper zigzag in which P 2i−1 ∈ C k+1 . Each zigzag P 0 , P 1 , ..., P 2m is a path connecting P 0 and P 2m , and so the distance on the zigzag d(P 0 , P 2m )=2m is always no less than the distance d S (P 0 , P 2m ) on an antimatroid (U, S ).
In Figure 6 we can see two sets (A = {1, 2, 3, 5} and B = {1, 3, 4, 5}) that are connected by an unique lower zigzag, such that the distance on the zigzag is 4, while |A △ B| = 2. Note, that the distance on the upper zigzag is indeed 2. For two sets X = {1, 2, 5} and Y = {3, 4, 5} both distances on the lower zigzag and on the upper zigzag are equal to 6, while |X △ Y| = 4.
In order to get the distance on a zigzag equal to the set distance, an antimatroid has to be a poset antimatroid. Assume that the hypothesis of the theorem is correct for all sets X ∈Swith |X| < k. Note that if for a zigzag P 0 , P 1 , ..., P 2m the distance on the zigzag d(P 0 , P 2m )=2m is equal to the distance d S (P 0 , P 2m )=|P 0 △ P 2m | then the zigzag preserves the distance for each pair P i , P j , i.e., d S (P i , P j )=d(P i , P j )=|j − i|.

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Poly-Dimension of Antimatroids For poset antimatroids there are distance preserving zigzags connecting two given sets, but these zigzags are not obliged to connect all feasible sets of the same cardinality. In Figure 8 we can see that there is a poset antimatroids, for which it is not possible to build two distance preserving zigzags connecting all feasible sets of the same cardinality. Fig. 8. A poset antimatroid without total zigzags.
It is worth mentioning that in 2-antimatroids all feasible sets of the same cardinality may be connected by one distance preserving zigzag.
Proposition 4.7. In 2-antimatroid (U, S ) all feasible sets C k of the same cardinality k can be connected both by a lower and an upper zigzag in such a way that the distance between any two sets d S (P i , P j ) belonging to the same zigzag coincides with distance on the zigzag d(P i , P j )=|j − i|.
Proof. We proceed by induction on cardinality k. For one-elements sets it is obvious. Assume that it is correct for all sets A ∈Swith |A| < k.
Each set from C k is obtained from some set of cardinality k − 1. Since an antimatroid is closed under union and its maximum in-degree and out-degree is 2, the n + 1 sets of cardinality k − 1 connected by an upper zigzag form n sets from C k . In addition to these n sets, each of two extreme sets (P 0 and P 2n ) can form one more set of cardinality k that does not belong to the upper zigzag. Since each out-degree is less than 3 there are no other sets of cardinality k.S o , the set C k is connected by a lower zigzag. Since an antimatroid is closed under union and its maximum in-degree is 2, all feasible sets of the same cardinality k are also connected by an upper zigzag.
Now prove that the distance on the zigzags coincides with the distance on the antimatroid. Since maximum in-degree and out-degree are 2, the obtained lower zigzag is only lower zigzag connecting P 0 and P 2n . Then from Theorem 4.6 it follows that it is a distance preserving zigzag. Similarly, we obtain that the upper zigzag preserves the distance as well.
For each edge (A, B) ∈Sthere exists a single element c ∈ B such that A = B − c,a n ds o we can label each edge by such element. Then, each 2-antimatroid consists of quadrilaterals with equal labels on opposite pairs of edges, since if it contains the vertices X, X ∪ a, X ∪ b for some X ∈S,anda, b ∈ U, it also contains X ∪{a, b}. This property, called by ∪-coloring, was invented in (Felsner & Knauer, 2009) as the characterization of join-distributive lattices.
Proposition 4.8. There exists a labeling of edges of a 2-antimatroid by two labels x or y such that opposite pairs of edges in each quadrilateral have equal labels.
Proof. To obtain such labeling we consequently scan a 2-antimatroid by layers C k beginning from the empty set (k = 0). The first zigzag of edges we mark by two labels x and y in turn, beginning from x or from y arbitrary. From the Proposition 4.7 follows that the next zigzag can be marked alternate by two labels (x, y) in such a way that the opposite pairs of edges in each quadrilateral have equal labels (see Figure 9).
If we reach the layer C k with one element, the next zigzag is labeled alternate by two labels without dependence on the previous layers. It follows from accessibility that for each A ∈Sthere is a shortest path connecting A with ∅.

①①①①①
The length of such path is equal to the cardinality of A.LetM(A) be a multiset containing all labels on this shortest path. Obviously, M(A) ∈ Z 2 . It is possible that there are some shortest paths between A and ∅. Prove that all such paths result in the same multiset.
Lemma 4.9. In every 2-antimatroid (U, S ) for each A ∈Sall shortest paths connecting A with ∅ forms the same multiset M(A).
Proof. We use induction on cardinality k. For one-elements sets it is obviously. Assume it is correct for all sets A ∈Swith |A| < k.C o n s i d e rY ∈ C k .I fY is obtained from only one set of cardinality k − 1, then M(Y) is unique. If Y = A ∪ a = B ∪ b, then there is X = A ∩ B ∈S such that A = X ∪ b and B = X ∪ a, since every 2-antimatroid is a poset antimatroid. Let a label of (X, X ∪ b) be x,andalabelof(X, X ∪ a) be y.T h u sM(A)=M(X) ∪ x and M(B)= M(X) ∪ y,a n ds oM(Y)=M(X) ∪{x, y}. Therefore, by induction hypothesis, all shortest path connecting Y with ∅ forms the same M(Y). The second edge (A 1 , A 2 ) is labeled by x and belongs to P A . The same is correct for each set on path P A from A till A ∩ B.S ot h ep a r to fP A from A till A ∩ B is labeled by label x only. By the same way we obtain that the part of P B from B till A ∩ B is labeled by y only.
Thus we have the following.
Theorem 4.11. The poly-dimension of a 2-antimatroid is two.