Two constructions of oriented matroids with disconnected extension space (original) (raw)
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Realizable but not Strongly Euclidean Oriented Matroids†
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Workshop Program Combinatorial geometries 2018 : matroids , oriented matroids and applications
2018
s of talks Monday 24 at 9h: A matroid extension result James OXLEY (Louisiana State University, USA) Let (A,B) be a 3-separation in a matroid M . If M is representable, then, in the underlying projective space, there is a line where the subspaces spanned by A and B meet, and M can be extended by adding elements from this line. In general, Geelen, Gerards, and Whittle proved that M can be extended by an independent set {p, q} such that {p, q} is in the closure of each of A and B. In this extension, each of p and q is freely placed on the line L spanned by {p, q}. This talk will discuss a result that gives necessary and su cient conditions under which a xed element can be placed on L. Monday 24 at 9h25: Su cient condition for almost irreducibility Csongor CSEHI (Budapest University of Technology and Economics, Hungary) A matroid N is almost irreducible if for any decomposition to matroid union M1 ∨M2 = N , N is a series extension of a submatroid of one of the matroids Mi ∈ {M1,M2}. An...
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Discrete & Computational Geometry, 2000
We provide a multiple purpose algorithm for generating oriented matroids. An application disproves a conjecture of Grünbaum that every closed triangulated orientable 2-manifold can be embedded geometrically in R 3 , i.e., with flat triangles and without selfintersections. We can show in particular that there exists an infinite class of orientable triangulated closed 2-manifolds for each genus g ≥ 6 that cannot be embedded geometrically in Euclidean 3-space. Our algorithm is interesting in its own right as a tool for many investigations in which oriented matroids play a key role.
The Topological Representation of Oriented Matroids
Discrete & Computational Geometry, 2005
We present a new direct proof of a topological representation theorem for oriented matroids in the general rank case. Our proof is based on an earlier rank 3 version. It uses hyperline sequences and the generalized Schönflies theorem. As an application, we show that one can read off oriented matroids from arrangements of embedded spheres of codimension one, even if wild spheres are involved.
A Topological Representation Theorem for Oriented Matroids
Discrete & Computational Geometry, 2002
We present a new direct proof of a topological representation theorem for oriented matroids in the general rank case. Our proof is based on an earlier rank 3 version. It uses hyperline sequences and the generalized Sch{\"o}nflies theorem. As an application, we show that one can read off oriented matroids from arrangements of embedded spheres of codimension one, even if
Extendable shellability for rank 3 matroid complexes
1994
We prove that matroid complexes of rank 3 are extendably shellable. Let (,") be the family of k-element subsets of a finite set E. There is a conjecture of Simon [lo] that (f) is extendably shellable for every k. In this paper we will prove that matroid complexes of rank 3 are extendably shellable. Simon's conjecture for k = 3 then follows as a special case, since (t) is the uniform matroid of rank 3. For BE(~) we will write B for the simplicial complex generated by B (i.e. such that B is the set of maximal faces of B). Our convention is that 0 E B. We will write B, /*B, if B1 GB, s(f) and B, shells to B,, i.e. a sequence of shelling steps extends B1 to BZ. By a shelling step is meant adding to BG(~) a set A@)-B such that A-B= {C: DGCGA} f or some D G A. A family B z(f) (or the simplicial complex B) is called shellable if 0 /* B, and extendably shellable if also 0 P C implies C /* B for all C E B.
A homotopy theorem on oriented matroids
Discrete Mathematics, 1993
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Polarity and point extensions in oriented matroids
Linear Algebra and its Applications, 1987
A. Bachem and W. Kern have recently extended the notion of polarity (relatively to an R-bilinear form) to oriented matroids [l]. We prove that the usual polarity properties of the face lattices of convex polytopes can be extended to the class of oriented matroids admitting an (oriented) polar. We give also a short proof of the principal result of [l] showing that there is a natural embedding of the poset of signed span of the cocircuits of a polar of an oriented matroid into the extension poset of this matroid. We remark that if M is a matroid admitting a polar, then every hyperplane can be intersected by every line. Oriented matroids satisfying this condition have an important role in oriented-matroid programming. 1. NOTATION We assume the reader is familiar with the basic results of oriented matroid theory [3, 8, 10, 111. We specify some of the notation used in this paper. A signed set X =(X', X-) is a finite set & = X+ U X-partitioned into two distinguished subsets: the set X+ of positive elements and the set X-of negative elements. The opposite-X of a signed set X is defined by (-X)' = X-and (-X)-= X+. For the sake of simplicity we use also the notation +X=X,andif Aisasubsetof& X-A=(X+-A,X-A)and XnA=(X+nA,X-nA), ,X=((X+-A)u(X-nA),
A characterization of uniform matroids
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arXiv: Combinatorics, 2019
We study Dressians of matroids using the initial matroids of Dress and Wenzel. These correspond to cells in regular matroid subdivisions of matroid polytopes. We characterize matroids that do not admit any proper matroid subdivisions. An efficient algorithm for computing Dressians is presented, and its implementation is applied to a range of interesting matroids.