For a matroid M, define the algebraic characteristic set x4(M) to be the set of field characteristics over which M can be algebraically represented. We construct many examples of rank three matroids with finite, non-singleton algebraic characteristic sets. We also determine x,,(PG(Z, p)) and xa(AG(2, p)). An infinite family of rank three matroids with empty algebraic characteristic set is const...

Let E be a possibly infinite set and let M N matroids defined on E. We say that the pair {M,N} has Intersection property if share an independent I admitting bipartition IM⊔IN such spanM(IM)∪spanN(IN)=E. The Matroid Conjecture of Nash-Williams says every matroid property. conjecture is known easy to prove in case when one uniform it was shown by Bowler Carmesin implied its special where direct s...

The number of disjoint cocircuits in a matroid is bounded by its rank. There are, however, matroids with arbitrarily large rank that do not contain two disjoint cocircuits; consider, for example, M(Kn) and Un,2n. Also the bicircular matroids B(Kn) have arbitrarily large rank and have no 3 disjoint cocircuits. We prove that for each k and n there exists a constant c such that, if M is a matroid ...

The number of disjoint co-circuits in a matroid is bounded by its rank. There are, however, matroids with arbitrarily large rank that do not contain two disjoint co-circuits; consider, for example, M(Kn) and Un,2n. Also the bicircular matroids B(Kn) have arbitrarily large rank and have no 3 disjoint co-circuits. We prove that for each k and n there exists a constant c such that, if M is a matro...

We study the lattice lat(M) of cocycles of a binary matroid M. By an isomorphism we show that such lattices are equivalent to lattices generated by the columns of proper submatrices of Sylvester matrices of full row length. As an application we show that the cocycle lattice of a recursively deened class of matroids, including all binary matroids of rank four, always has a basis consisting of co...

In this paper we look at complexity aspects of the following problem (matroid representability) which seems to play an important role in structural matroid theory: Given a rational matrix representing the matroid M , the question is whether M can be represented also over another specific finite field. We prove this problem is hard, and so is the related problem of minor testing in rational matr...

This article presents a complete proof of Mnëv’s Universality Theorem and and a first complete proof of Mnëv’s Universal Partition Theorem for oriented matroids. The Universality Theorem states that, for every primary semialgebraic set V there is an oriented matroid M, whose realization space is stably equivalent to V . The Universal Partition Theorem states that, for every partition V of IR in...

An element e of a 3-connected matroid M is essential if neither the deletion M\e nor the contraction M/e 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 this paper, we consider those 3-connected matroids that have some non-essential elements, showing that every such matroid M must have ...

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...