Matrix Theory for Independence Algebras
Document Type
Article
Digital Object Identifier (DOI)
10.1016/j.laa.2022.02.021
Journal Title
Linear Algebra and its Applications
Publication Date
2022
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, Glazek 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.
Publisher Statement
This article is freely available through Elsevier's ScienceDirect.
Recommended Citation
Araújo, João, Wolfram Bentz, Peter J. Cameron, Michael Kinyon, and Janusz Konieczny. “Matrix Theory for Independence Algebras.” Linear Algebra and Its Applications 642 (June 2022): 221–50. https://doi.org/10.1016/j.laa.2022.02.021.