Matrix theory for independence algebras
Abstract
A universal algebra 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 set X of can be extended to an endomorphism of . 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 be any independence algebra of finite dimension n, with at least two elements. Denote by End() the monoid of endomorphisms of . 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. 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 Szabo in 2000. (This answers another question by experts on independence algebras.) We also extend the classification of Cameron and Szabo 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.
Citation
Araujo , J , Bentz , W , Cameron , P J , Kinyon , M & Konieczny , J 2022 , ' Matrix theory for independence algebras ' , Linear Algebra and its Applications , vol. 642 , pp. 221-250 . https://doi.org/10.1016/j.laa.2022.02.021
Publication
Linear Algebra and its Applications
Status
Peer reviewed
ISSN
0024-3795Type
Journal article
Description
Funding: This work was funded by national funds through the FCT - Fundação para a Ciência e a Tecnologia, I.P., under the scope of the projects UIDB/00297/2020, UIDP/00297/2020 (Center for Mathematics and Applications) and PTDC/MAT/PUR/31174/2017.Collections
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