Monoids over which products of indecomposable acts are indecomposable

TL;DR

This paper characterizes monoids based on the indecomposability of products and subacts of indecomposable acts, linking these properties to the existence of zero elements and reversibility in the monoid.

Contribution

It provides necessary and sufficient conditions for when products and subacts of indecomposable acts are indecomposable, connecting these to zero elements and reversibility in monoids.

Findings

Products of indecomposable acts are indecomposable iff the monoid has a right zero.

Subacts of indecomposable acts are indecomposable iff the monoid is left reversible.

The trivial act is product flat iff the monoid has a left zero.

Abstract

In this paper we prove that for a monoid $S$, products of indecomposable right $S$-acts are indecomposable if and only if $S$ contains a right zero. Besides, we prove that subacts of indecomposable right $S$-acts are indecomposable if and only if $S$ is left reversible. Ultimately, we prove that the one element right $S$-act $\Theta_S$ is product flat if and only if $S$ contains a left zero.