Matrix theory for independence algebras

Abstract

A universal algebra A with underlying set A is said to be a matroid algebra if (A,〈⋅〉), where 〈⋅〉 denotes the operator subalgebra generated by, is a matroid. A matroid algebra is said to be an independence algebra if every mapping α:X→A defined on a minimal generating X of A can be extended to an endomorphism of A. These algebras are particularly well-behaved generalizations of vector spaces, and hence they naturally appear in several branches of mathematics, such as model theory, group theory, and semigroup theory.It is well known that matroid algebras have a well-defined notion of dimension. Let A be any independence algebra of finite dimension n, with at least two elements. Denote by End(A) the monoid of endomorphisms of A. In the 1970s, Głazek proposed the problem of extending the matrix theory for vector spaces to a class of universal algebras which included independence algebras. In this paper, we answer that problem by developing a theory of matrices for (almost all) finite-dimensional independence algebras.In the process of solving this, we explain the relation between the classification of independence algebras obtained by Urbanik in the 1960s, and the classification of finite independence algebras up to endomorphism-equivalence obtained by Cameron and Szabó in 2000. (This answers another question by experts on independence algebras.) We also extend the classification of Cameron and Szabó to all independence algebras.The paper closes with a number of questions for experts on matrix theory, groups, semigroups, universal algebra, set theory or model theory.