Weakly right coherent monoids

TL;DR

This paper investigates the properties of weakly right coherent monoids, focusing on their algebraic structure, closure properties under certain constructions, and connections to other algebraic concepts like right ideal Howson and flatness.

Contribution

It extends the understanding of weakly right coherent monoids by analyzing their behavior under algebraic constructions and exploring their relation to right ideal Howson and flatness.

Findings

Closure results for weakly right coherent monoids under algebraic constructions

Characterization of right ideal Howson property in terms of axiomatisability

Connections between right ideal Howson and flatness conditions

Abstract

A monoid $S$ is said to be weakly right coherent if every finitely generated right ideal of $S$ is finitely presented as a right $S$-act. It is known that $S$ is weakly right coherent if and only if it satisfies the following conditions: $S$ is right ideal Howson, meaning that the intersection of any two finitely generated right ideals of $S$ is finitely generated; and the right annihilator congruences of $S$ are finitely generated as right congruences. We examine the behaviour of these two conditions (in the more general setting of semigroups) under certain algebraic constructions and deduce closure results for the class of weakly right coherent monoids. We also show that the property of being right ideal Howson is related to the axiomatisability of a class of left acts satisfying a condition related to flatness.