We prove that the maximum size of a simple binary matroid of rank r ≥ 5 with no AG(3, 2)-minor is (r+1 2 ) and characterize those matroids achieving this bound. When r ≥ 6, the graphic matroid M(Kr+1) is the unique matroid meeting the bound, but there are a handful of matroids of lower ranks meeting or exceeding this bound. In addition, we determine the size function for nongraphic simple binar...

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...

Let Mn be a linear hyperplane arrangement in IR. We define finite posets Gk(M) and Vk(M) of oriented matroids associated with this, which approximate the Grassmannian Gk(IR) and the Stiefel manifold Vk(IR), respectively. The basic conjectures are that the “OM-Grassmannian” Gk(M) has the homotopy type of Gk(IR), and that the “OM-Stiefel bundle” ∆π : ∆Vk(M) −→ ∆Gk(M) is a surjective map. These co...

The generalization of binary operation in the classical algebra to fuzzy is an important development field algebra. paper proposes a new vector spaces over field, which called M-hazy field. Some fundamental properties spaces, and subspaces are studied, some results also proved. Furthermore, linear transformation studied their Finally, it shown that M-fuzzifying convex induced by subspace space.

A graft is a representation of an even cut matroid M if the cycles of M correspond to the even cuts of the graft. Two, long standing, open questions regarding even cut matroids are the problem of finding an excluded minor characterization and the problem of efficiently recognizing this class of matroids. Progress on these problems has been hampered by the fact that even cut matroids can have an...

We introduce “matroid parse trees” which, using only a limited amount of information at each node, can build up the vector representations of matroids of bounded branch-width over a finite field. We prove that if M is a family of matroids described by a sentence in the monadic second-order logic of matroids, then there is a finite tree automaton accepting exactly those parse trees which build v...

The main content of the note is a proof of the conjecture of Hamidoune-Las Vergnas on the directed switching game in the case of Lawrence oriented matroids. C.E. Shannon has introduced the switching game for graphs circa 1960. It has been generalized and solved for matroids by A. Lehman [4]. A switching game on graphs and oriented matroids was introduced by Y. O. Hamidoune and M. Las Vergnas [3...

Based on a complete Heyting algebra, we modify the definition oflattice-valued fuzzifying convergence space using fuzzy inclusionorder and construct in this way a Cartesian-closed category, calledthe category of $L-$ordered fuzzifying convergence spaces, in whichthe category of $L-$fuzzifying topological spaces can be embedded.In addition, two new categories are introduced, which are called the...

in this paper the concepts of fuzzifying β − irresolute functions and fuzzifying β − compactspaces are characterized in terms of fuzzifying β − open sets and some of their properties are discussed.

This paper is based on the element splitting operation for binary matroids that was introduced by Azadi as a natural generalization of the corresponding operation in graphs. In this paper, we consider the problem of determining precisely which graphic matroids M have the property that the element splitting operation, by every pair of elements on M yields a graphic matroid. This problem is solve...