We characterize all internally 4-connected binary matroids M with the property that the ground set of M can be ordered (e0, . . . , en−1) in such a way that {ei, . . . , ei+t} is 4-separating for all 0 ≤ i, t ≤ n − 1 (all subscripts are read modulo n). We prove that in this case either n ≤ 7 or, up to duality, M is isomorphic to the polygon matroid of a cubic or quartic planar ladder, the polyg...

Given a matroid M with distinguished element e, a port oracle with respect to e reports whether or not a given subset contains a circuit that contains e. The rst main result of this paper is an algorithm for computing an e-based ear decomposition (that is, an ear decomposition every circuit of which contains element e) of a matroid using only a polynomial number of elementary operations and por...

A structure M is pregeometric if the algebraic closure is a pregeometry in all M ′ elementarily equivalent to M . We define a generalisation: structures with an existential matroid. The main examples are superstable groups of U-rank a power of ω and d-minimal expansion of fields. Ultraproducts of pregeometric structures expanding a field, while not pregeometric in general, do have an unique exi...

Let F4 be the root system associated with the 24-cell, and let M(F4) be the simple linear dependence matroid corresponding to this root system. We determine the automorphism group of this matroid and compare it to the Coxeter group W for the root system. We find non-geometric automorphisms that preserve the matroid but not the root system.

Tropical varieties play an important role in algebraic geometry. The Bergman complex B(M) and the positive Bergman complex B+(M) of an oriented matroid M generalize to matroids the notions of the tropical variety and positive tropical variety associated to a linear ideal. Our main result is that if A is a Coxeter arrangement of type Φ with corresponding oriented matroid MΦ, then B (MΦ) is dual ...

A new so-called fuzzifying measurable theory that generalizes the classical measurable theory is established and the structures of such new theory are discussed very detailed. In the last, we study the product of two fuzzifying measures and consider a problem, which is like the third of the open problems in fuzzy measure presented by Z. Wang. We have solved this problem satisfactorily in the ne...

In this paper we consider the problem of finding the densest subset subject to co-matroid constraints. We are given a monotone supermodular set function f defined over a universe U , and the density of a subset S is defined to be f(S)/|S|. This generalizes the concept of graph density. Co-matroid constraints are the following: given matroid M a set S is feasible, iff the complement of S is inde...

It is well known that a matroid is binary if and only if it has no minor isomorphic to U2,4, the 4-point line. Extending this result, Bixby proved that every element in a nonbinary connected matroid is in a U2,4minor. The result was further extended by Seymour who showed that every pair of elements in a nonbinary 3-connected matroid is in a U2,4-minor. This paper extends Seymour's theorem by pr...

For matroids M and N on disjoint sets S and T , a semidirect sum of M and N is any matroid K on S ∪ T that, like the direct sum and the free product, has the restrictionK|S equal toM and the contractionK/S equal toN . We abstract a matrix construction to get a general matroid construction: the matroid union of any rank-preserving extension ofM on the set S ∪ T with the direct sum ofN and the ra...

Let M be a class of matroids representable over a field F. A matroid N # M stabilizes M if, for any 3-connected matroid M # M, an F-representation of M is uniquely determined by a representation of any one of its N-minors. One of the main theorems of this paper proves that if M is minor-closed and closed under duals, and N is 3-connected, then to show that N is a stabilizer it suffices to check...