Some Results on Factorization of Monoids

TL;DR

This paper investigates the conditions under which monoids can be factorized into submonoids, providing classifications and exploring connections with non-abelian cohomology and semi-direct products.

Contribution

It introduces necessary and sufficient conditions for monoid factorizations using descent 1-cocycles and classifies factorizations involving subgroups, linking to non-abelian cohomology.

Findings

Characterization of monoid factorizations via descent 1-cocycles

Complete classification of factorizations with subgroup factors

Analysis of monoid factorizations in relation to non-abelian cohomology

Abstract

Factorizations of monoids are studied. Two necessary and sufficient conditions in terms of so-called descent 1-cocyles for a monoid to be factorized through two submonoids are found. A full classification of those factorizations of a monoid whose one factor is a subgroup of the monoid is obtained. The relationship between monoid factorizations and non-abelian cohomology of monoids is analyzed. Some applications to semi-direct product of monoids are given.