Related papers: Oriented Matroid Theory as a Mathematical Framewor…
The theory of matroids has been generalized to oriented matroids and, recently, to arithmetic matroids. We want to give a definition of "oriented arithmetic matroid" and prove some properties like the "uniqueness of orientation".
A link between matroid theory and $p$-branes is discussed. The Schild type action for $p$-branes and matroid bundle notion provide the two central structures for such a link. We use such a connection to bring the duality concept in matroid…
We argue that the chirotope concept of oriented matroid theory may be found in different scenarios of physics, including classical mechanics, quantum mechanics, gauge field theory, p-branes formalism, two time physics and Matrix theory. Our…
We assume gravity in a $d$-dimensional manifold $M$ and consider a splitting of the form $M=M_{p}\times M_{q}$, with $d=p+q$. The most general two-block metric associated with $M_{p}$ and $M_{q}$ is used to derive the corresponding…
We introduce a construction of oriented matroids from a triangulation of a product of two simplices. For this, we use the structure of such a triangulation in terms of polyhedral matching fields. The oriented matroid is composed of…
We considered the possibility that the oriented matroid theory is connected with supersymmetry via the Grassmann-Plucker relations. The main reason for this, is that such relations arise in both in the chirotopes definition of an oriented…
In this paper we present a definition of oriented Lagrangian symplectic matroids and their representations. Classical concepts of orientation and this extension may both be thought of as stratifications of thin Schubert cells into unions of…
The phirotope is a complex generalization of the concept of chirotope in oriented matroid theory. Our main goal in this work is to establish a link between phirotopes, super p-branes and qubit theory. For this purpose we first discuss…
We present an algebraic framework which simultaneously generalizes the notion of linear subspaces, matroids, valuated matroids, and oriented matroids. We call the resulting objects matroids over hyperfields. In fact, there are (at least)…
In this paper we extend the theory of oriented matroids to Lagrangian orthogonal matroids and their representations, and give a completely natural transformation from a representation of a classical oriented matroid to a representation of…
We study the combinatorial properties of a tropical hyperplane arrangement. We define tropical oriented matroids, and prove that they share many of the properties of ordinary oriented matroids. We show that a tropical oriented matroid…
Matroids and semigraphoids are discrete structures abstracting and generalizing linear independence among vectors and conditional independence among random variables, respectively. Despite the different nature of conditional independence…
This paper defines the q-analogue of a matroid and establishes several properties like duality, restriction and contraction. We discuss possible ways to define a q-matroid, and why they are (not) cryptomorphic. Also, we explain the…
We study the exceptional U duality group $E_d$ of M-theory compactified on a d-torus and its representations using Matrix theory. We exhibit the $E_d$ structure and show that p-branes wrapped or unwrapped around the longitudinal direction…
We explore a combinatorial theory of linear dependency in complex space, "complex matroids", with foundations analogous to those for oriented matroids. We give multiple equivalent axiomatizations of complex matroids, showing that this…
The separation theorem of Kirchberger can be proven using a combination of Farkas' Lemma and Caratheodory's Theorem. Since those theorems are at the heart of oriented matroids, we are interested in a generalization of Kirchberger's Theorem…
A transduction provides us with a way of using the monadic second-order language of a structure to make statements about a derived structure. Any transduction induces a relation on the set of these structures. This article presents a…
A matroid is a machine capturing linearity of mathematical objects and producing combinatorial structures. Matroid structure arises everywhere since linearity is a ubiquitous concept. One natural way to obtain matroids is by considering…
A mixed graph is a graph with some directed edges and some undirected edges. We introduce the notion of mixed matroids as a generalization of mixed graphs. A mixed matroid can be viewed as an oriented matroid in which the signs over a fixed…
Is it possible to define cryptomorphic axiom systems for infinite oriented matroids by lifting some of the axiom systems for finite oriented matroids to the infinite setting while not losing duality in the process? We show that the answer…