The determinant of finite semigroups of the pseudovariety ECOM

TL;DR

This paper computes the non-zero semigroup determinant for finite semigroups where all idempotents commute, extending previous classes and impacting coding theory over semigroup algebras.

Contribution

It provides the first explicit computation of the determinant for this broader class of semigroups, advancing algebraic coding theory.

Findings

Computed the non-zero semigroup determinant for the class of semigroups with commuting idempotents.

Extended the class of semigroups for which the determinant is known beyond inverse semigroups.

Implications for extending the MacWilliams theorem in coding theory.

Abstract

The purpose of this paper is to compute the non-zero semigroup determinant of the class of finite semigroups in which every two idempotents commute. This class strictly contains the class of finite semigroups that have central idempotents and the class of finite inverse semigroups. This computation holds significance in the context of the extension of the MacWilliams theorem for codes over semigroup algebras.