A step to compute the determinant of finite semigroups not in ECom

TL;DR

This paper explores methods to compute the nonzero determinant of finite semigroups with non-commutative idempotents, specifically focusing on a class called -smooth semigroups, relevant for coding theory extensions.

Contribution

It introduces the concept of -smooth semigroups and studies their determinant computation, advancing understanding in algebraic coding theory.

Findings

Defined -smooth semigroups.

Established methods for determinant computation in this class.

Potential applications to extended MacWilliams theorem.

Abstract

The purpose of this paper is to begin studying the computation of the nonzero determinant of semigroups within the class of finite semigroups that possesses a pair of non-commutative idempotents. This paper focuses on a class of these semigroups introduced as $\ll$-smooth semigroups. This computation is applicable in the context of the extension of the MacWilliams theorem for codes over semigroup algebras.