scholarly journals Topological representation of dual pairs of oriented matroids

1993 ◽  
Vol 10 (3) ◽  
pp. 237-240 ◽  
Author(s):  
Thomas H. Brylawski ◽  
Günter M. Ziegler
Keyword(s):  
Oriented Matroids ◽  
Topological Representation ◽  
Dual Pairs

scholarly journals On the Combinatorial Structure of Arrangements of Oriented Pseudocircles

10.37236/1783 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Johann Linhart ◽  
Ronald Ortner
Keyword(s):  
Representation Theorem ◽  
Equivalence Classes ◽  
Oriented Matroids ◽  
Combinatorial Structure ◽  
Topological Representation ◽  
Combinatorial Properties ◽  
One To One ◽  
Orientable Surfaces

We introduce intersection schemes (a generalization of uniform oriented matroids of rank 3) to describe the combinatorial properties of arrangements of pseudocircles in the plane and on closed orientable surfaces. Similar to the Folkman-Lawrence topological representation theorem for oriented matroids we show that there is a one-to-one correspondence between intersection schemes and equivalence classes of arrangements of pseudocircles. Furthermore, we consider arrangements where the pseudocircles separate the surface into two components. For these strict arrangements there is a one-to-one correspondence to a quite natural subclass of consistent intersection schemes.

scholarly journals Tropical Oriented Matroids

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Silke Horn
Keyword(s):  
Representation Theorem ◽  
Oriented Matroids ◽  
Oriented Matroid ◽  
Topological Representation ◽  
International Audience

International audience Tropical oriented matroids were defined by Ardila and Develin in 2007. They are a tropical analogue of classical oriented matroids in the sense that they encode the properties of the types of points in an arrangement of tropical hyperplanes – in much the same way as the covectors of (classical) oriented matroids describe the types in arrangements of linear hyperplanes. Not every oriented matroid can be realised by an arrangement of linear hyperplanes though. The famous Topological Representation Theorem by Folkman and Lawrence, however, states that every oriented matroid can be represented as an arrangement of $\textit{pseudo}$hyperplanes. Ardila and Develin proved that tropical oriented matroids can be represented as mixed subdivisions of dilated simplices. In this paper I prove that this correspondence is a bijection. Moreover, I present a tropical analogue for the Topological Representation Theorem. La notion de matroïde orientè tropical a été introduite par Ardila et Develin en 2007. Ils sont un analogue des matroïdes orientés classiques dans le sens où ils codent les propriétés des types de points dans un arrangement d'hyperplans tropicaux – d'une manière très similaire à celle dont les covecteurs des matroïdes orientés (classiques) décrivent les types de points dans les arrangements d'hyperplans linéaires. Tous les matroïdes orientés ne peuvent pas être représentés par un arrangement d'hyperplans linéaires. Cependant le célèbre théorème de représentation topologique de Folkman et Lawrence affirme que tout matroïde orientè peut être représenté par un arrangement de $\textit{pseudo}$-hyperplans. Ardila et Develin ont prouvè que les matroïdes orientés tropicaux peuvent être représentés par des sous-divisions mixtes de simplexes dilatés. Je prouve dans cet article que cette correspondance est une bijection. Je présente en outre, un analogue tropical du théorème de représentation topologique.

scholarly journals The Topological Representation of Oriented Matroids

2005 ◽  
Vol 33 (4) ◽  
pp. 645-668 ◽  
Author(s):  
J�rgen Bokowski ◽  
Simon King ◽  
Susanne Mock ◽  
Ileana Streinu
Keyword(s):  
Oriented Matroids ◽  
Topological Representation

scholarly journals CHOICE-FREE STONE DUALITY

2019 ◽  
Vol 85 (1) ◽  
pp. 109-148
Author(s):  
NICK BEZHANISHVILI ◽  
WESLEY H. HOLLIDAY
Keyword(s):  
Boolean Algebra ◽  
Boolean Algebras ◽  
Stone Space ◽  
Spectral Space ◽  
Topological Representation ◽  
Stone Duality ◽  
Closed Sets ◽  
Spectral Spaces ◽  
Basic Facts ◽  
Regular Open

AbstractThe standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this article, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone used in his representation of distributive lattices via compact open sets. It also takes advantage of Tarski’s observation that the regular open sets of any topological space form a Boolean algebra. We prove without choice principles that any Boolean algebra arises from a special spectral space X via the compact regular open sets of X; these sets may also be described as those that are both compact open in X and regular open in the upset topology of the specialization order of X, allowing one to apply to an arbitrary Boolean algebra simple reasoning about regular opens of a separative poset. Our representation is therefore a mix of Stone and Tarski, with the two connected by Vietoris: the relevant spectral spaces also arise as the hyperspace of nonempty closed sets of a Stone space endowed with the upper Vietoris topology. This connection makes clear the relation between our point-set topological approach to choice-free Stone duality, which may be called the hyperspace approach, and a point-free approach to choice-free Stone duality using Stone locales. Unlike Stone’s representation of Boolean algebras via Stone spaces, our choice-free topological representation of Boolean algebras does not show that every Boolean algebra can be represented as a field of sets; but like Stone’s representation, it provides the benefit of a topological perspective on Boolean algebras, only now without choice. In addition to representation, we establish a choice-free dual equivalence between the category of Boolean algebras with Boolean homomorphisms and a subcategory of the category of spectral spaces with spectral maps. We show how this duality can be used to prove some basic facts about Boolean algebras.

Explicit Solvability of Dual Pairs of Infinite Linear Programs

OPSEARCH ◽  
2005 ◽  
Vol 42 (3) ◽  
pp. 288-296
Author(s):  
K. C. Sivakumar ◽  
J. Mercy Swarna
Keyword(s):  
Linear Programs ◽  
Dual Pairs ◽  
Infinite Linear
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