Let $E$ be a finite set and $\mathcal P$, $\mathcal S$, $\mathcal L$ three classes of subsets of $E$, and $r$ a function defined on $2^E$. In this paper, we give an algorithm for testing if the quadruple $(\mathcal P, \mathcal S, \mathcal L, r)$ is the locked structure of a given matroid, i.e., recognizing if $(\mathcal P, \mathcal S, \mathcal L, r)$ defines a matroid. This problem is intractab...

Matroid theory is a generalization of the idea of linear independence. A matroid M consists of a finite set E (called the ground set) and a collection S of subsets of E satsifying the following conditions: (1) ∅ ∈ S; (2) if I ∈ S, then every subset of I is in S; (3) if I1 and I2 are in S and |I1| < |I2|, then there is an element e of I2 − I1 such that I1 ∪ e is in S. The elements of S are calle...

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...

The independent sets of an algebraic matroid are sets of algebraically independent transcendentals over a field k. If a matroid M is isomorphic to an algebraic matroid the latter is called an algebraic representation of M. Vector representations of matroids are defined similarly. A matroid may have algebraic (resp. vector) representations over fields of different characteristics. The problem in...

Given a linear space L in affine space A, we study its closure L̃ in the product of projective lines (P). We show that the degree, multigraded Betti numbers, defining equations, and universal Gröbner basis of its defining ideal I(L̃) are all combinatorially determined by the matroid M of L. We also prove I(L̃) and all of its initial ideals are Cohen-Macaulay with the same Betti numbers, and can be...

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...

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 is a survey of algorithmic results in the theory of “discrete convex analysis” for integer-valued functions defined on integer lattice points. The theory parallels the ordinary convex analysis, covering discrete analogues of the fundamental concepts such as conjugacy, the Fenchel min-max duality, and separation theorems. The technical development is based on matroid-theoretic concepts, in ...