Matrix theory for independence algebras

Joao Araujo; Wolfram Bentz; Peter J. Cameron; Michael Kinyon; Janusz  Konieczny

doi:10.1016/j.laa.2022.02.021

Matrix theory for independence algebras

Joao Araujo^*, Wolfram Bentz, Peter J. Cameron, Michael Kinyon, Janusz Konieczny

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

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 set 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.

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.

Original language	English
Pages (from-to)	221-250
Number of pages	30
Journal	Linear Algebra and its Applications
Volume	642
Early online date	2 Mar 2022
DOIs	https://doi.org/10.1016/j.laa.2022.02.021
Publication status	Published - 1 Jun 2022

Keywords

Universal algebra
Matroid
Semigroups
Matrix theory
Groups
Fields
Model theory

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10.1016/j.laa.2022.02.021

Araujo-2022-Matrix-theory-LAA-AAM
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title = "Matrix theory for independence algebras",

abstract = "A universal algebra 픸 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 픸 can be extended to an endomorphism of 픸. These algebras are particularly well-behavedgeneralizations 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 픸 be any independence algebra of finite dimension n, with at least two elements. Denote by End(픸) the monoid of endomorphisms of 픸. 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.",

keywords = "Universal algebra, Matroid, Semigroups, Matrix theory, Groups, Fields, Model theory",

author = "Joao Araujo and Wolfram Bentz and Cameron, {Peter J.} and Michael Kinyon and Janusz Konieczny",

note = "Funding: This work was funded by national funds through the FCT - Funda{\c c}{\~a}o para a Ci{\^e}ncia 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.",

year = "2022",

month = jun,

day = "1",

doi = "10.1016/j.laa.2022.02.021",

language = "English",

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pages = "221--250",

journal = "Linear Algebra and its Applications",

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T1 - Matrix theory for independence algebras

AU - Araujo, Joao

AU - Bentz, Wolfram

AU - Cameron, Peter J.

AU - Kinyon, Michael

AU - Konieczny, Janusz

N1 - 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.

PY - 2022/6/1

Y1 - 2022/6/1

N2 - A universal algebra 픸 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 픸 can be extended to an endomorphism of 픸. These algebras are particularly well-behavedgeneralizations 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 픸 be any independence algebra of finite dimension n, with at least two elements. Denote by End(픸) the monoid of endomorphisms of 픸. 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.

AB - A universal algebra 픸 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 픸 can be extended to an endomorphism of 픸. These algebras are particularly well-behavedgeneralizations 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 픸 be any independence algebra of finite dimension n, with at least two elements. Denote by End(픸) the monoid of endomorphisms of 픸. 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.

KW - Universal algebra

KW - Matroid

KW - Semigroups

KW - Matrix theory

KW - Groups

KW - Fields

KW - Model theory

U2 - 10.1016/j.laa.2022.02.021

DO - 10.1016/j.laa.2022.02.021

M3 - Article

SN - 0024-3795

VL - 642

SP - 221

EP - 250

JO - Linear Algebra and its Applications

JF - Linear Algebra and its Applications

ER -

Matrix theory for independence algebras

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