Matroids from hypersimplex splits

Matroids from hypersimplex splits

The Graph of the Hypersimplex

The (k, d)-hypersimplex is a (d − 1)-dimensional polytope whose vertices are the (0, 1)vectors that sum to k. When k = 1, we get a simplex whose graph is the complete graph Kd. Here we show how many of the well known graph parameters and attributes of Kd extend to a more general case. In particular we obtain explicit formulas in terms of d and k for the number of vertices, vertex degree, number...

CATEGORICAL RELATIONS AMONG MATROIDS, FUZZY MATROIDS AND FUZZIFYING MATROIDS

The aim of this paper is to study the categorical relations betweenmatroids, Goetschel-Voxman’s fuzzy matroids and Shi’s fuzzifying matroids.It is shown that the category of fuzzifying matroids is isomorphic to that ofclosed fuzzy matroids and the latter is concretely coreflective in the categoryof fuzzy matroids. The category of matroids can be embedded in that offuzzifying matroids as a simul...

Matroids from modules

The aim of this work is to show that (oriented) matroid methods can be applied to many discrete geometries, namely those based on modules over integral (ordered) domains. The trick is to emulate the structure of a vector space within the module, thereby allowing matroid methods to be used as if the module were a vector space. Only those submodules which are “closed under existing divisors,” and...

From Clutters to Matroids

This paper deals with the question of completing a monotone increasing family of subsets Γ of a finite set Ω to obtain the dependent sets of a matroid. Specifically, we provide several natural processes for transforming the clutter Λ of the inclusionminimal subsets of the family Γ into the set of circuits C(M) of a matroid M with ground set Ω. In addition, by combining these processes, we prove...