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... Arnaldo Mandel gave in his dissertation [13] the following conjecture. We call oriented matroids fulfilling that conjecture Mandel. ...
... Proof. In [13], Chapter VI, Proposition 5, page 320, this is the equivalence from (5.1) to (5.4). The proof of the equivalences was left as an exercise there. ...
... The fact, that Euclideaness implies the existence of simplicial topes in an oriented matroid was already mentioned in Mandels dissertation [13] and in [10], Theorem 4.5. We reformulate this result in another way. ...
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We call an oriented matroid Mandel if it has an extension in general position which makes all programs with that extension Euclidean. If L is the minimum number of mutations adjacent to an element of the groundset, we call an oriented matroid Las Vergnas if L>0L > 0. If Oproperty\frak{O}_{\mathcal{property}} is the class of oriented matroids having a certain property, it holds OOLasVergnasOMandelOEuclideanOrealizable.\frak{O} \supset \frak{O}_{\mathcal{Las Vergnas}} \supset \frak{O}_{\mathcal{Mandel}} \supset \frak{O}_{\mathcal{Euclidean}} \supset \frak{O}_{\mathcal{realizable}}. All these inclusions are proper, we give explicit proofs/examples for the parts of this chain that were not known. For realizable hyperplane arrangements of rank r we have L=rL = r which was proved by Shannon. Under the assumption that a (modified) intersection property holds we give an analogon to Shannons proof and show that uniform rank 4 Euclidean oriented matroids with that property have L=4L = 4. Using the fact that the lexicographic extension creates and destroys certain mutations, we show that for Euclidean oriented matroids holds L3L \ge 3. We give a survey of preservation of Euclideaness and prove that Euclideaness remains after a certain type of mutation-flips. This yields that a path in the mutation graph from a Euclidean oriented matroid to a totally non-Euclidean oriented matroid (which has no Euclidean oriented matroid programs) must have at least three mutation-flips. Finally, a minimal non-Euclidean or rank 4 uniform oriented matroid is Mandel if it is connected to a Euclidean oriented matroid via one mutation-flip, hence we get many examples for Non-Euclidean but Mandel oriented matroids and have L3L \le 3 for those of rank 4.
... Topological representation: A cornerstone in the theory of oriented matroids is the topological representation theorem by Folkman and Lawrence [47] which states that every oriented matroid has a representation by a pseudosphere arrangement and allows us to consider oriented matroids as topological objects. A later proof based on piecewise linear topology is given by Edmonds and Man- del [75]. Bohne and Dress [17, 41] revealed the connection of zonotopal tilings with oriented matroids. ...
... Although the " wiggly " pseudoarrangements still maintain a lot of the combinatorial and topological properties of hyperplane arrangements they are quite hard to handle without additional restrictions to make them " nice " . For example Edmonds and Mandel used the concept of piecewise linear topology [75] in order to maintain some control on the possible behaviour in arrangements of pseudohemispheres, which were introduced by Folkman and Lawrence [47] as a topological representation of the combinatorial concept of an oriented matroid. ...
... The following notion of arrangements of pseudo(hemi)spheres was introduced by Folkman and Lawrence in [47] and much simplified by Edmonds and Mandel in [75]. In condition (A1), if S A ? S e = S ?1 = ? ...
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An arrangement is a collection of subspaces of a topological space. For example, a set of codimension one affine subspaces in a finite dimensional vector space is an arrangement of hyperplanes. A general question in arrangement theory is to determine to what extent the combinatorial data of an arrangement determines the topology of the complement of the arrangement. Established combinatorial structures in this context are matroids and -for hyperplane arrangements in the real vector space- oriented matroids. Let X be the punctured plane C- 0 or the unit circle S 1, and a(1),...,a(n) integer vectors in Z d. By interpreting the a(i) as characters of the torus T=Hom(Z d,X) isomorphic to X d we obtain a toric arrangement in T by considering the set of kernels of the characters. A toric arrangement is covered naturally by a periodic affine hyperplane arrangement in the d-dimensional complex or real vector space V=C d or R d (according to whether X = C- 0 or S 1). Moreover, if V is the real vector space R d the stratification of V given by a finite hyperplane arrangement can be combinatorially characterized by an affine oriented matroid. Our main objective is to find an abstract combinatorial description for the stratification of T given by the toric arrangement in the case X=S 1 - and to develop a concept of toric oriented matroids as an abstract characterization of arrangements of topological subtori in the compact torus (S 1) d. Part of our motivation comes from the possible generalization of known topological results about the complement of "complexified" toric arrangements to such toric pseudoarrangements. Towards this goal, we study abstract combinatorial descriptions of locally finite hyperplane arrangements and group actions thereon. First, we generalize the theory of semimatroids and geometric semilattices to the case of an infinite ground set, and study their quotients under group actions from an enumerative and structural point of view. As a second step, we consider corresponding generalizations of affine oriented matroids in order to characterize the stratification of R d given by a locally finite non-central arrangement in R d in terms of sign vectors.
... A tope [8,11] of an oriented matroid is a maximal element of its cocircuit span. The notion of topes has a natural meaning in typical examples of oriented matroids and has also been studied under the name of 'non-Radon partitions' (see [2,4,5,9]). ...
... 41 0195-6698/90/010041 + 05 $02.00/0 © 1989 Academic Press Limited by its topes (see [4]). For the set fY, the following three properties are well known (see [3,6,8,11]): ...
... The property (T3) is the essential property for the proof of the shellability of tope cells in oriented matroid (see [3,4,6,8,11]). ...
Article
A tope of an oriented matroid is a maximal element in its cocircuit span. In this note we give a simple characterization of oriented matroids in terms of topes. Using this we answer an open problem of Edelman by giving a characterization of oriented matroids in terms of acyclic sets.
... More precisely, let R denote the collection of regions of A: the (closures of) connected components of the complement of A in R n . For any two regions C and C ′ , sep(C, C ′ ) is the set of hyperplanes H ∈ A that separate C and C ′ ; that is, such that C and C ′ are not contained in the same halfspace bounded by H. Fix a base region B ∈ R. The weak order with base region B [Man82,Ede84] is the partial order relation ⪯ B on R defined by ...
... The definitions and results of this section hold for arbitrary linear arrangements, including nonsimplicial arrangements. Then, ⪯ B is a partial order with minimum element B and maximum element B, the region opposite to the base region B. This order is called the weak order of A with base region B, and was independently introduced by Mandel [Man82] and Edelman [Ede84]. Mandel's definition was given in the context of oriented matroids, where it was called the Tope graph of A. This order was further studied by Björner, Edelman and Ziegler [BEZ90], who showed that whenever A is simplicial, the weak order with respect to any base region is a lattice. ...
... More precisely, let R denote the collection of regions of A: the (closures of) connected components of the complement of A in R n . For any two regions C and C ′ , sep(C, C ′ ) is the set of hyperplanes H ∈ A that separate C and C ′ ; that is, such that C and C ′ are not contained in the same halfspace bounded by H. Fix a base region B ∈ R. The weak order with base region B [17,33] is the partial order relation ⪯ B on R defined by ...
... The definitions and results of this section hold for arbitrary linear arrangements, including non-simplicial arrangements. Then, ⪯ B is a partial order with minimum element B and maximum element B, the region opposite to the base region B. This order is called the weak order of A with base region B, and was independently introduced by Mandel [33] and Edelman [17]. Mandel's definition was given in the context of oriented matroids, where it was called the Tope graph of A. This order was further studied by Björner, Edelman and Ziegler [7], who showed that whenever A is simplicial, the weak order with respect to any base region is a lattice. ...
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We introduce the Primitive Eulerian polynomial PA(z)P_{\cal A}(z) of a central hyperplane arrangement A{\cal A}. It is a reparametrization of its cocharacteristic polynomial. Previous work of the first author implicitly show that, for simplicial arrangements, PA(z)P_{\cal A}(z) has nonnegative coefficients. For reflection arrangements of type A and B, the same work interprets the coefficients of PA(z)P_{\cal A}(z) using the (flag)excedance statistic on (signed) permutations. The main result of this article is to provide an interpretation of the coefficients of PA(z)P_{\cal A}(z) for all simplicial arrangements only using the geometry and combinatorics of A{\cal A}. This new interpretation sheds more light to the case of reflection arrangements and, for the first time, gives combinatorial meaning to the coefficients of the Primitive Eulerian polynomial of the reflection arrangement of type D. In type B, we find a connection between the Primitive Eulerian polynomial and the 1/2-Eulerian polynomial of Savage and Viswanathan (2012). We present some real-rootedness results and conjectures for PA(z)P_{\cal A}(z).
... After Hansen's counterexample for the natural generalization of Sylvester's theorem to higher dimension, another generalization was proven by him in [11] where he proved that any real point configuration of any dimension has a simple hyperplane but not necessarily an independent one. Later on Murty [14] conjectured that Hansen's theorem is also true for oriented matroids i.e. any simple oriented matroid (matroid without loops or parallels) has a simple hyperplane. ...
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In 1993, Csima and Sawyer proved that in a non-pencil arrangement of n pseudolines, there are at least 613n\frac{6}{13}n simple points of intersection. Since pseudoline arrangements are the topological representations of reorientation classes of oriented matroids of rank 3, in this paper, we will use this result to prove by induction that an oriented paving matroid of rank r3r \ge 3 on n elements, where n5+rn \geq 5+ r, has at least 1213(r1)(nr2)\frac{12}{13(r-1)} \binom{n}{r-2} independent hyperplanes, yielding a new necessary condition for a paving matroid to be orientable.
... and Folkman & Lawrence [20], and further investigated by Edmonds & Mandel [19] and many other authors , oriented matroids represent a unified combinatorial theory of orientations of ordinary matroids, which simultaneously captures the basic properties of sign vectors representing the regions in a hyperplane arrangement in R n and of sign vectors of the circuits in a directed graph. Furthermore, oriented matroids find applications in point and vector configurations, convex polytopes, and linear programming. ...
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In his seminal 1983 paper, Jim Lawrence introduced lopsided sets and featured them as asymmetric counterparts of oriented matroids, both sharing the key property of strong elimination. Moreover, symmetry of faces holds in both structures as well as in the so-called affine oriented matroids. These two fundamental properties (formulated for covectors) together lead to the natural notion of "conditional oriented matroid" (abbreviated COM). These novel structures can be characterized in terms of three cocircuits axioms, generalizing the familiar characterization for oriented matroids. We describe a binary composition scheme by which every COM can successively be erected as a certain complex of oriented matroids, in essentially the same way as a lopsided set can be glued together from its maximal hypercube faces. A realizable COM is represented by a hyperplane arrangement restricted to an open convex set. Among these are the examples formed by linear extensions of ordered sets, generalizing the oriented matroids corresponding to the permutohedra. Relaxing realizability to local realizability, we capture a wider class of combinatorial objects: we show that non-positively curved Coxeter zonotopal complexes give rise to locally realizable COMs.
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