The present paper investigates the relations between fuzzifying topologies and generalized ideals of fuzzy subsets, as well as constructing generalized ideals and fuzzifying topologies by means of fuzzy preorders. Furthermore, a construction of generalized ideals from preideals, and vice versa, is obtained. As a particular consequence of the results in this paper, a construction of fuzzifying t...

The aim of this paper is to give insight into the behaviour of inequivalent representations of 3{connected matroids. An element x of a matroid M is xed if there is no extension M 0 of M by an element x 0 such that fx;x 0 g is independent and M 0 is unaltered by swapping the labels on x and x 0. When x is xed, a representation of M nx extends in at most one way to a representation of M. A 3{conn...

In this paper, we first extend the concept of arity in crisp convex spaces to case fuzzification and give some related properties. From view hull operator, study relations between disjoint sum M-fuzzifying its factor spaces. We also examine additivity degree separability (S0,S1,S2,S3,S4). Finally, show that every space is JHC iff corresponding JHC.

Knowing the excluded minors for a minor-closed matroid property provides a useful alternative characterization of that property. It has been shown in [R. Hall, J. Oxley, C. Semple, G. Whittle, On Matroids of Branch-Width Three, submitted 2001] that if M is an excluded minor for matroids of branch-width 3, then M has at most 14 elements. We show that there are exactly 10 such binary matroids M (...

The main purpose of this paper is to introduce a concept of L-fuzzifying topological vector spaces (here L is a completely distributive lattice) and study some of their basic properties. Also, a characterization of such spaces in terms of the corresponding L-fuzzifying neighborhood structure of the zero element is given. Finally, the conclusion that the category of L-fuzzifying topological vect...

We extend the complexity theoretic framework of reachability problems in graphs to the case of matroids. Given a matroid M and two elements s and t of the ground set, the reachability problem is to test if there is a circuit in M containing both s and t. We show complexity characterizations for several important classes of matroids. In particular, we show: (1) For two important classes of matro...

It has been conjectured that asymptotically almost all matroids are sparse paving, i.e. that s(n) ∼ m(n), where m(n) denotes the number of matroids on a fixed groundset of size n, and s(n) the number of sparse paving matroids. In an earlier paper, we showed that log s(n) ∼ logm(n). The bounds that we used for that result were dominated by matroids of rank r ≈ n/2. In this paper we consider the ...

This paper introduces a general matroid-theoretic construction which includes, as special cases, elementary lifts of matroids and bias matroids of biased graphs. To perform the construction on a matroid A/ , it is necessary (but not sufficient) to have a submodular function inducing M . Elementary lifts are obtained when the submodular function chosen is the rank function of M . We define what ...

We begin by giving some background to this result. A matroid M is an excluded minor for a minor-closed class of matroids if M is not in the class but all proper minors of M are. It is natural to attempt to characterize a minor-closed class of matroids by giving a complete list of its excluded minors. Unfortunately, a minor-closed class of matroids can have an infinite number of excluded minors....

provided again that M is simple and binary. The reason for restricting attention here to simple binary matroids is that expression (1.1) need not hold for arbitrary simple matroids or even for loopless graphic matroids. To see, this, consider, for example, the uniform matroids U 2 ,m form~ 6 and U 1,n for n ~ 4. We observe here that inequality (1.3) is a sharper bound than expression (1.2) unle...