On the Folkman–Lawrence Topological Representation Theorem for Oriented Matroids of Rank 3

Jürgen Bokowski; Susanne Mock; Ileana Streinu

doi:10.1006/eujc.2000.0482

On the Folkman–Lawrence Topological Representation Theorem for Oriented Matroids of Rank 3

European Journal of Combinatorics ◽

10.1006/eujc.2000.0482 ◽

2001 ◽

Vol 22 (5) ◽

pp. 601-615 ◽

Cited By ~ 6

Author(s):

Jürgen Bokowski ◽

Susanne Mock ◽

Ileana Streinu

Keyword(s):

Representation Theorem ◽

Oriented Matroids ◽

Topological Representation

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A Topological Representation Theorem for tropical oriented matroids

Journal of Combinatorial Theory Series A ◽

10.1016/j.jcta.2016.03.003 ◽

2016 ◽

Vol 142 ◽

pp. 77-112 ◽

Cited By ~ 3

Author(s):

Silke Horn

Keyword(s):

Representation Theorem ◽

Oriented Matroids ◽

Topological Representation

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On the Combinatorial Structure of Arrangements of Oriented Pseudocircles

The Electronic Journal of Combinatorics ◽

10.37236/1783 ◽

2004 ◽

Vol 11 (1) ◽

Cited By ~ 2

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.

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Tropical Oriented Matroids

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.3026 ◽

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.

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