The determinant of \(\lll\)-smooth semigroups

TL;DR

This paper studies determinants of a special class of finite semigroups called ll-smooth semigroups, developing methods for their computation with applications to coding theory and extending previous work on semigroup determinants.

Contribution

It introduces ll-smooth semigroups and develops a new method for computing their determinants, extending prior approaches and enabling applications in coding theory.

Findings

Developed a method for computing contracted semigroup determinants.

Extended the approach from previous work to ll-smooth semigroups.

Potential applications in extending the MacWilliams theorem for codes.

Abstract

This paper continues the investigation of non-zero determinants associated with finite semigroups containing a pair of non-commuting idempotents, as initiated in~\cite{Sha-Det2}. We focus on a class of semigroups, called \( \lll \)-smooth semigroups, that allow meaningful structural analysis despite the absence of \( \ll \)-transitivity. Within this framework, we develop a method for computing contracted semigroup determinants, building on and extending the approach carried over from~\cite{Sha-Det2}. These computations are motivated by applications in coding theory, particularly by the potential extending the MacWilliams theorem for codes over semigroup algebras.