The largest subsemilattices of the endomorphism monoid of an independence algebra

Araújo, João; Bentz, Wolfram; Konieczny, Janusz

Abstract

An algebra A is said to be an independence algebra if it is a matroid algebra and every map α:X→A, defined on a basis 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. We prove that a largest subsemilattice of End(A) has either 2n-1elements (if the clone of A does not contain any constant operations) or 2nelements (if the clone of A contains constant operations). As corollaries, we obtain formulas for the size of the largest subsemilattices of: some variants of the monoid of linear operators of a finite-dimensional vector space, the monoid of full transformations on a finite set X, the monoid of partial transformations on X, the monoid of endomorphisms of a free G-set with a finite set of free generators, among others. The paper ends with a relatively large number of problems that might attract attention of experts in linear algebra, ring theory, extremal combinatorics, group theory, semigroup theory, universal algebraic geometry, and universal algebra. © 2014 Elsevier Inc.

Citation

Araújo, J., Bentz, W., & Konieczny, J. (2014). The largest subsemilattices of the endomorphism monoid of an independence algebra. Linear algebra and its applications, 458, 60-79. https://doi.org/10.1016/j.laa.2014.05.041

• Journal Article Type: Article
• Acceptance Date: May 27, 2014
• Online Publication Date: Jun 20, 2014
• Publication Date: Oct 1, 2014
• Deposit Date: Jun 29, 2018
• Journal: Linear Algebra and its Applications
• Print ISSN: 0024-3795
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 458
• Pages: 60-79
• DOI: https://doi.org/10.1016/j.laa.2014.05.041
• Keywords: Independence algebra; Semilattice; Monoid of endomorphisms; Dimension
• Public URL: https://hull-repository.worktribe.com/output/901212
• Publisher URL: https://www.sciencedirect.com/science/article/pii/S0024379514003619?via%3Dihub
• Additional Information: This article is maintained by: Elsevier; Article Title: The largest subsemilattices of the endomorphism monoid of an independence algebra; Journal Title: Linear Algebra and its Applications; CrossRef DOI link to publisher maintained version: http://dx.doi.org/10.1016/j.laa.2014.05.041; Content Type: article; Copyright: Copyright © 2014 Elsevier Inc. All rights reserved.