TY - JOUR

T1 - Matrix theory for independence algebras

AU - Araújo, João

AU - Bentz, Wolfram

AU - Cameron, Peter J.

AU - Kinyon, Michael

AU - Konieczny, Janusz

N1 - Funding Information:
info:eu-repo/grantAgreement/FCT/6817 - DCRRNI ID/UIDB%2F00297%2F2020/PT#
info:eu-repo/grantAgreement/FCT/6817 - DCRRNI ID/UIDP%2F00297%2F2020/PT#
PTDC/MAT/PUR/31174/2017
Publisher Copyright:
© 2022 Elsevier Inc.

PY - 2022/6/1

Y1 - 2022/6/1

N2 - 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.

AB - 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.

KW - Fields

KW - Groups

KW - Matrix theory

KW - Model theory

KW - Semigroups

KW - Universal algebra

UR - http://www.scopus.com/inward/record.url?scp=85125518516&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2022.02.021

DO - 10.1016/j.laa.2022.02.021

M3 - Article

AN - SCOPUS:85125518516

SN - 0024-3795

VL - 642

SP - 221

EP - 250

JO - Linear Algebra and its Applications

JF - Linear Algebra and its Applications

ER -