Finitely Presented Monoids and Algebras defined by Permutation Relations of Abelian Type

TL;DR

This paper studies finitely presented algebras and monoids defined by permutation relations of abelian type, proving the Jacobson radical is zero and characterizing cancellativity based on stabilizers in the permutation group.

Contribution

It extends previous work by characterizing the Jacobson radical and cancellativity conditions for these algebras and monoids with abelian permutation relations.

Findings

Jacobson radical of the algebras is zero

Cancellativity characterized by stabilizers in the permutation group

Provides conditions for algebra and monoid properties

Abstract

The class of finitely presented algebras over a field K with a set of generators a_1,...,a_n and defined by homogeneous relations of the form a_1a_2...a_n = a_{sigma(1)}a_{sigma(2)}...a_{sigma(n)}, where sigma runs through an abelian subgroup H of Sym_{n}, the symmetric group, is considered. It is proved that the Jacobson radical of such algebras is zero. Also, it is characterized when the monoid S_n(H), with the "same" presentation as the algebra, is cancellative in terms of the stabilizer of 1 and the stabilizer of n in H. This work is a continuation of earlier work of Cedo, Jespers and Okninski.