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.

One of the most basic examples of matroid duality is the following. Let G be a graph imbedded in the plane and let G∗ be its topological dual graph. If M(G) is the cycle matroid of G, then the dual matroid M∗(G) = M(G∗). If G is a connected graph that is 2-cell imbedded in a surface of demigenus d > 0 (the demigenus is equal to 2 minus the euler characteristic of the surface), then M∗(G) 6= M(G...

The independent assignment problem (or the weighted matroid intersection problem) is extended using Dress-Wenzel’s matroid valuations, which are attached to the vertex set of the underlying bipartite graph as an additional weighting. Specifically, the problem considered is: Given a bipartite graph G = (V , V −;A) with arc weight w : A → R and matroid valuations ω and ω− on V + and V − respectiv...

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...

D.W. Hall proved that every simple 3-connected graph with a Ks-minor must have a K3.3-minor, the only exception being Ks itself. In this paper, we prove that every 3-connected binary matroid with an M(Ks)-minor must have an M(K3.3)or M*(K3,3)-minor, the only exceptions being M(K5), a highly symmetric 12-element matroid which we call T12, and any single-element contraction of T~2.

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.

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...

We give a counterexample to a conjecture by Wild about binary matroids. We connect two equivalent lines of research in matroid theory: a simple type of basis-exchange property and restrictions on the cardinalities of intersections of circuits and cocircuits. Finally, we characterize direct sums of series-parallel networks by a simple basis-exchange property. In [9], Wild proposed a characteriza...

This paper proves a preliminary step towards a splitter theorem for internally 4-connected binary matroids. In particular, we show that, provided M or M∗ is not a cubic Möbius or planar ladder or a certain coextension thereof, an internally 4-connected binary matroid M with an internally 4-connected proper minor N either has a proper internally 4-connected minor M ′ with an N -minor such that |...

Let M and N be ternary matroids having the same rank and the same ground set, and assume that every independent set in N is also independent in M . The main result of this paper proves that if M is 3-connected and N is connected and non-binary, then M = N . A related result characterizes precisely when a matroid that is obtained by relaxing a circuit-hyperplane of a ternary matroid is also tern...