ing from the behavior of linearly independent sets of columns of a matrix, in 1935, Whitney defined a matroid M to consist of a finite set E (or E(M)) and a collection I (or I(M)) of subsets of E called independent sets with the properties that the empty set is independent; every subset of an independent set is independent; and if one independent set has more elements than another, then an elem...

This article studies the girth and cogirth problems for a connected matroid. The problem of finding the cogirth of a graphic matroid has been intensively studied, but studies on the equivalent problem for a vector matroid or a general matroid have been rarely reported. Based on the duality and connectivity of a matroid, we prove properties associated with the girth and cogirth of a matroid whos...

In this paper, the theory of γ -convergence and γ c -convergence on nets and filters is established in fuzzifying topology. Some important and interesting results in fuzzifying topology are obtained by means of the theory.

A series class P of a connected matroid M is removable if M \ P is connected. In this paper, we prove that a connected matroid M with r(M) ≥ 2 has at least r(M) + 1 removable series classes. Further, we obtain certain results from which the following result of Oxley and its graph theoretic version follow: If C is a circuit of a connected matroid M with C 6= M such that M \ x is not connected fo...

A 3-separation (A, B), in a matroid M, is called sequential if the elements of A can be ordered (a1 , ..., ak) such that, for i=3, ..., k, ([a1 , ..., a i], [ai+1 , ..., ak] _ B) is a 3-separation. A matroid M is sequentially 4-connected if M is 3-connected and, for every 3-separation (A, B) of M, either (A, B) or (B, A) is sequential. We prove that, if M is a sequentially 4-connected matroid t...

Let AC = {H1, . . . , Hn} be a (central) arrangement of hyperplanes in C. Let M(AC) be the dependence matroid of the linear forms {θHi ∈ (C ) : Ker(θHi ) = Hi}. The Orlik-Solomon algebra OS(M) of a matroid M is the exterior algebra on the points modulo the ideal generated by circuit boundaries. The algebra OS(M(AC)) is isomorphic to the cohomology algebra of the manifold M = C \ ⋃ H∈AC H. The T...

The concepts of fuzzy e-open sets and fuzzy e-continuous are introduced and studied in fuzzifying topology and by making use of these concepts, we introduce and study T e 0 -, R e 0-, T e 1 -, R e 1-, T e 2 -, T e 3 -, T e 4 -, strong T e 3 and strong T e 4 separation axioms in fuzzifying topology and give some of their characterizations as well as the relations of these axioms and other separa...

We prove that, for each positive integer k, every sufficiently large 3-connected regular matroid has a parallel minor isomorphic to M∗(K3,k), M(Wk), M(Kk), the cycle matroid of the graph obtained from K2,k by adding paths through the vertices of each vertex class, or the cycle matroid of the graph obtained from K3,k by adding a complete graph on the vertex class with three vertices.

Welsh conjectured that for any simple regular connected matroid M, if each cocircuit has at least 1 2 (r(M) + 1) elements, then there is a circuit of size r(M) + 1. This conjecture was proven by Hochstättler and Jackson in 1997. In this paper, we give a shorter proof of this conjecture based solely on matroid-theoretical arguments. Let M be a simple, connected, regular matroid and let C ∈ C (M)...

A nongraphic matroid M is said to be almost-graphic if, for all elements e, either M\e or M/e is graphic. We determine completely the class of almost-graphic matroids, thereby answering a question posed by Oxley in his book “Matroid Theory.” A nonregular matroid is said to be almost-regular if, for all elements e, either M\e or M/e is regular. An element e for which both M\e and M/e are regular...