A transversal matroid M can be represented by a collection of sets, called a presentation of M , whose partial transversals are the independent sets of M . Minimal presentations are those for which removing any element from any set gives a presentation of a different matroid. We study the connections between (single-element) transversal extensions of M and extensions of presentations of M . We ...

The results c ited in the title are unifi ed by th e following theore m: For a ny matroid M a nd any subse ts Nand K of e le me nts in M , there e xi s t as many as k disjoint subsets of N which span K a nd which spa n each ot he r if and only if the re is no contraction matroid M X A where NnA partitions int o as few as k se ts such th at eac h is independen t in M X A and suc h that at least ...

The connections between algebra and finite geometry are very old, with theorems about configurations of points dating to ancient Greece. In these notes, we will put a matroid theoretic spin on these results, with matroid representations playing the central role. Recall the definition of a matroid via independent sets I. Definition 1.1. Let E be a finite set and let I be a family of subsets of E...

Given a group $G$ of automorphisms matroid $M$, we describe the representations on homology independence complex dual $M^*$. These are related with lattice flats and (when $M$ is realizable) top cohomology hyperplane arrangement. Finally analyze in detail case complete graph, which has applications to algebraic geometry.

the main purpose of this paper is to introduce a concept of$l$-fuzzifying topological groups (here $l$ is a completelydistributive lattice) and discuss some of their basic properties andthe structures. we prove that its corresponding $l$-fuzzifyingneighborhood structure is translation invariant. a characterizationof such topological groups in terms of the corresponding$l$-fuzzifying neighborhoo...

Given a root system R, the vector system R̃ is obtained by taking a representative v in each antipodal pair {v,−v}. The matroid M(R) is formed by all independent subsets of R̃. The automorphism group of a matroid is the group of permutations preserving its independent subsets. We prove that the automorphism groups of all irreducible root systems matroids M(R) are uniquely determined by their inde...

Based on the optimality criteria established in Part I, we show a primal-type cyclecanceling algorithm and a primal-dual-type augmenting algorithm for the valuated independent assignment problem: Given a bipartite graph G = (V , V −;A) with arc weight w : A → R and matroid valuations ω and ω− on V + and V − respectively, find a matching M(⊆ A) that maximizes ∑ {w(a) | a ∈ M}+ ω(∂M) + ω−(∂−M), w...

In the classical model of cooperative games, it is considered that each coalition of players can form and cooperate to obtain its worth. However, we can think that in some situations this assumption is not real, that is, all the coalitions are not feasible. This suggests that it is necessary to rise the whole question of generalizing the concept of cooperative game, and therefore to introduce a...

Given a linear space L in affine space A, we study its closure L̃ in the product of projective lines (P). We show that the degree, multigraded Betti numbers, defining equations, and universal Gröbner basis of its defining ideal I(L̃) are all combinatorially determined by the matroid M of L. We also prove I(L̃) and all of its initial ideals are Cohen-Macaulay with the same Betti numbers, and can be...

Introduction In this note I show how very general and powerful results about the union and intersection of matroids due to J. Edmonds [19] may be deduced from a matroid generalisation of Hall's theorem by R. Rado [13]. Throughout, S, T, will denote finite sets, |X| will denote the cardinality of the set X and {xt: iel} denotes the set whose distinct elements are the elements x{. A matroid (S, M...