Oriented matroids can serve as a tool of modeling of collective decision-making processes in contradictory problems of pattern recognition. We present a generalization of the committee techniques of pattern recognition to oriented matroids. A tope committee for an oriented matroid is a subset of its maximal covectors such that every positive halfspace contains more than half of the maximal covectors from this subset. For a large subfamily of oriented matroids their committee structure is quite rich; for example, any maximal chains in their tope posets provide one with information sufficient to construct a committee.
A three-tope committee K* for a simple oriented matroid M is a 3-subset of its maximal covectors such that every positive halfspace of M contains at least two topes from K*. We consider three-tope committees as the vertex sets of triangles in graphs associated with the topes and enumerate them making use of the properties of the poset of convex subsets of the ground set of M.
The principle of inclusion-exclusion is applied to subsets of maximal covectors contained in halfspaces of a simple oriented matroid and to convex subsets of its ground set for enumerating tope committees.
Let the sign components of the maximal covectors of a simple oriented matroid M be represented by the real numbers -1 and 1. Consider the vertex set V(R) of a symmetric cycle R of adjacent topes in the tope graph of M as a subposet of the tope poset of M. If B is the bottom element of the tope poset then B is equal to the unweighted sum of the members of the set min V(R) of minimal elements of the subposet V(R); if B is the positive tope then the set min V(R) is a critical tope committee for the acyclic oriented matroid M.
A tope committee K* for a simple oriented matroid M is a subset of its maximal covectors such that every positive halfspace of M contains more than half of the covectors from K*. The structures of the family of all committees for M, and of the family of its committees that contain no pairs of opposites, are described. A Farey subsequence associated with the elements of the m-th layer of the Boolean lattice of rank 2m is explored.
For a symmetric 2t-cycle in the tope graph of a simple oriented matroid M on the ground set {1,...,t}, where t is even, we describe decompositions of topes and subtopes of M with respect to the subtopes corresponding to the edges of the symmetric cycle.
If V(R) is the vertex set of a symmetric cycle R in the tope graph of a simple oriented matroid M, then for any tope T of M there exists a unique inclusion-minimal subset Q(T;R) of V(R) such that T is the sum of the topes of Q(T;R). If |Q(T;R)|>3, then the decomposition Q(T;R) of the tope T with respect to the symmetric cycle R satisfies certain Dehn-Sommerville type relations.
If V(R) is the vertex set of a symmetric cycle R in the tope graph of a simple oriented matroid M, then for any tope T of M there exists a unique inclusion-minimal subset Q(T,R) of V(R) such that T is the sum of the topes of Q(T,R). If for decompositions Q(T',R') and Q(T",R") with respect to symmetric cycles R' and R" in the tope graphs of two simple oriented matroids, whose ground sets have the cardinalities of opposite parity, we have |Q(T',R')|>3 and |Q(T",R")|>3, then these decompositions satisfy a certain orthogonality relation.
If V(R) is the vertex sequence of a symmetric cycle R in the tope graph of a simple acyclic oriented matroid M on a t-element ground set, then the set min V(R) of minimal elements in the subposet V(R) of the tope poset of M, based at the positive tope, is a critical committee for M that votes for the base tope. We consider the sequence zR of poset ranks of the elements from the vertex sequence of R as a fragment of a signal with period 2t and relate the number of members of the committee min V(R) to the magnitudes of [t/2] components, with odd indices, of the discrete Fourier transform of the …
We give two graph theoretical characterizations of tope graphs of (complexes of) oriented matroids. The first is in terms of excluded partial cube minors, the second is that all antipodal subgraphs are gated. A direct consequence is a third characterization in terms of zone graphs of tope graphs. Further corollaries include a characterization of topes of oriented matroids due to da Silva, another one of Handa, a characterization of lopsided systems due to Lawrence, and an intrinsic characterization of tope graphs of affine oriented matroids. Furthermore, we obtain polynomial time recognition a…
We consider decompositions of topes of the oriented matroid realizable as the arrangement of coordinate hyperplanes in $\mathbb{R}^{2^t}$, with respect to a distinguished symmetric $2\cdot 2^t$-cycle in its hypercube graph of topes $\boldsymbol{H}(2^t,2)$. We seek interpretations of such decompositions in the context of subset families on the ground set $E_t:=\{1,\ldots,t\}$ and of the families of their blocking sets, in the context of clutters on $E_t$ and of their blockers.
In this paper we consider the question of the existence of Hamiltonian circuits in the tope graphs of central arrangements of hyperplanes. Some of the results describe connections between the existence of Hamiltonian circuits in the arrangement and the odd-even invariant of the arrangement. In conjunction with this, we present some results concerning bounds on the odd-even invariant. The results given here can be formulated more generally for oriented matroids and are still valid in that setting.
The components of K*-vectors associated to a simple oriented matroid M are the numbers of general or special tope committees for M. Using the principle of inclusion-exclusion, we determine how the reorientations of M on one-element subsets of its ground set affect K*-vectors.
Building on a recent characterization of tope graphs of Complexes of Oriented Matroids (COMs), we tackle and generalize several classical problems in Oriented Matroids (OMs), Lopsided Sets (aka ample set systems), and partial cubes via Metric Graph Theory. Our first main result is that every element of an OM from a class introduced by Mandel is incident to a simplicial tope, i.e, such OMs contain no mutation-free elements. This allows us to refute a conjecture of Mandel from 1983, that would have implied the famous Las Vergnas' simplex conjecture. Further, we show that the mutation graph of un…
We generalize the Varchenko matrix of a hyperplane arrangement to oriented matroids. We show that the celebrated determinant formula for the Varchenko matrix, first proved by Varchenko, generalizes to oriented matroids. It follows that the determinant only depends on the matroid underlying the oriented matroid and analogous formulas hold for cones in oriented matroids. We follow a proof strategy for the original Varchenko formula first suggested by Denham and Hanlon. Besides several technical lemmas this strategy also requires a topological result on supertopes which is of independent interest…
Varchenko introduced in 1993 a distance function on the chambers of a hyperplane arrangement that gave rise to a determinant whose entry in position $(C, D)$ is the distance between the chambers $C$ and $D$, and computed that determinant. In 2017, Aguiar and Mahajan provided a generalization of that distance function, and computed the corresponding determinant. This article extends their distance function to the topes of an oriented matroid, and computes the determinant thus defined. Oriented matroids have the nice property to be abstractions of some mathematical structures including hyperplan…
We introduce the notion of real phase structure on rational polyhedral fans in Euclidean space. Such a structure consists of an assignment of affine spaces over $\mathbb{Z}/2\mathbb{Z}$ to each top dimensional face of the fan subject to two conditions. Given an oriented matroid we can construct a real phase structure on the fan of the underlying matroid. Conversely, we show that from a real phase structure on a matroid fan we can produce an orientation of the underlying matroid. Thus real phase structures are cryptomorphic to matroid orientations. The topes of the orientated matroid are recove…
We define a partial ordering on the set Q = Q(M) of pairs of topes of an oriented matroid M, and show the geometric realization |Q| of the order complex of Q has the same homotopy type as the Salvetti complex of M. For any element e of the ground set, the complex |Qe| associated to the rank-one oriented matroid on {e} has the homotopy type of the circle. There is a natural free simplicial action of Z4 on |Q|, with orbit space isomorphic to the order complex of the poset Q(M,e) associated to the pointed (or affine) oriented matroid (M,e). If M is the oriented matroid of an arrangement A of line…
Neither simpliciality nor mutation connectivity: conjectures of Las Vergnas and Cordovil-Las Vergnas fail
University of Chicago（芝加哥大学）；Departament de Matemàtiques i Informàtica, Universitat de Barcelona（巴塞罗那大学数学与计算机系）；Centre de Recerca Matemàtica (CRM)（数学研究中心）
Department of Mathematics, Creighton University（克里顿大学数学系）；Department of Mathematics and Statistics, Boston University（波士顿大学数学与统计系）；Department of Mathematics, Florida Atlantic University（佛罗里达大西洋大学数学系）
