Congruences on direct products of transformation and matrix monoids

Araújo, João; Bentz, Wolfram; Gomes, Gracinda M. S.

Abstract

Malcev described the congruences of the monoid Tn of all full transformations on a finite set Xn={1,…,n}. Since then, congruences have been characterized in various other monoids of (partial) transformations on Xn, such as the symmetric inverse monoid Inn of all injective partial transformations, or the monoid PTn of all partial transformations. The first aim of this paper is to describe the congruences of the direct products Qm×Pn, where Q and P belong to {T,PT,In}. Malcev also provided a similar description of the congruences on the multiplicative monoid Fn of all n×n matrices with entries in a field F, our second aim is provide a description of the principal congruences of Fm×Fn. The paper finishes with some comments on the congruences of products of more than two transformation semigroups, and a fairly large number of open problems.

Citation

Araújo, J., Bentz, W., & Gomes, G. M. S. (2018). Congruences on direct products of transformation and matrix monoids. Semigroup Forum, 97(3), 384–416. https://doi.org/10.1007/s00233-018-9931-8

• Journal Article Type: Article
• Acceptance Date: Mar 5, 2018
• Online Publication Date: Apr 2, 2018
• Publication Date: 2018-12
• Deposit Date: Mar 18, 2018
• Publicly Available Date: Apr 3, 2019
• Print ISSN: 0037-1912
• Publisher: Springer Verlag
• Peer Reviewed: Peer Reviewed
• Volume: 97
• Issue: 3
• Pages: 384–416
• DOI: https://doi.org/10.1007/s00233-018-9931-8
• Keywords: Algebra and Number Theory
• Public URL: https://hull-repository.worktribe.com/output/746533
• Publisher URL: https://link.springer.com/article/10.1007%2Fs00233-018-9931-8
• Contract Date: Apr 9, 2018