Linear semigroups with permutation properties
A semigroup S satisfies a permutation property if for some \(n\geq 2\) every product of n elements of S remains invariant under some nontrivial permutation of its factors. It is known that the semigroup algebra K[S] of a finitely generated semigroup satisfying a permutation property over a field K satisfies a polynomial identity whenever S is periodic [\textit{A. Restivo} and \textit{C. Reutenauer}, J. Algebra 89, 102-104 (1984; Zbl 0545.20051)], or S is cancellative [\textit{J. Okniński}, Commun. Algebra 15, 1667-1677 (1987; Zbl 0621.20040)]. In the note under review it is shown that the same holds if S is a regular linear finitely generated semigroup satisfying a permutation property and char K\(=0.\) Reviewer's note. 1. The hypothesis on the characteristic of K is superfluous. 2. In forthcoming papers by the reviewer the same is shown for the class of inverse semigroups and a counterexample is found in the class of regular semigroups.
scientific article; zbMATH DE number 4015019