An essential element of a 3–connected matroid M is one for which neither the deletion nor the contraction is 3–connected. Tutte’s Wheels and Whirls Theorem proves that the only 3–connected matroids in which every element is essential are the wheels and whirls. In an earlier paper, the authors showed that a 3–connected matroid with at least one non-essential element has at least two such element...

Matroids are combinatorial structures that generalize the notion of linear independence in matrices. There are many equivalent definitions of matroids, we will use one that focus on its independent sets. A matroid M is defined on a finite ground set E (or E(M) if we want to emphasize the matroid M) and a collection of subsets of E are said to be independent. The family of independent sets is de...

Matroids are combinatorial structures that generalize the notion of linear independence in matrices. There are many equivalent definitions of matroids, we will use one that focus on its independent sets. A matroid M is defined on a finite ground set E (or E(M) if we want to emphasize the matroid M) and a collection of subsets of E are said to be independent. The family of independent sets is de...

In this paper,  it is shown that the category of $L$-ordered fuzzifying convergence spaces contains the category of pretopological $L$-ordered fuzzifying convergence spaces as a bireflective subcategory and the latter contains the category of topological $L$-ordered fuzzifying convergence spaces as a bireflective subcategory. Also, it is proved that the category of $L$-ordered fuzzifying conver...

Two elements in an oriented matroid are inseparable if they have either the same sign in every signed circuit containing them both or opposite signs in every signed circuit containing them both. Two elements of a matroid are adjacent if there is no M(K4)-minor using them both, and in which they correspond to a matching of K4. We prove that two elements e, { of an oriented matroid are inseparabl...

Let M be a matroid. When M is 3-connected, Tutte’s WheelsandWhirls Theorem proves that M has a 3-connected proper minor N with exactly one element fewer than M unless M is a wheel or a whirl. I will present a corresponding result for internally 4-connected binary matroids. This presentation is based on joint work by myself, Dillon Mayhew, and James Oxley.

In this paper, some characterizations of fuzzifying strong compactness are given, including in terms nets and presubbases. Several locally the framework topology introduced mapping theorems obtained.

This thesis deals with questions relating to the maximum density of rank-n matroids in a minor-closed class. Consider a minor-closed class M of matroids that does not contain a given rank-2 uniform matroid. The growth rate function is defined by hM(n) = max (|M | : M ∈M simple, r(M) ≤ n) . The Growth Rate Theorem, due to Geelen, Kabell, Kung, and Whittle, shows that the growth rate function is ...

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...

In this paper we determine completely the class of binary matroids with no minors isomor-phic to the cycle matroid of the prism graph M*(Ks\e), its dual M(Ks\e), and the binary affine cube AG(3, 2).