Weakly right coherent monoids
Levent Michael Dasar, Victoria Gould, Craig Miller
TL;DR
This work analyzes weakly right coherent monoids, defined by the finiteness of presentations of finitely generated right ideals as right acts, equivalently via right ideal Howson and finitely generated right annihilator congruences. It develops a comprehensive closure map for three interrelated notions—right ideal Howson (RIH), finitely right equated (FRE), and weakly right coherent (WRC)—under key semigroup constructions, including Rees quotients, retracts, direct products and free products, in both semigroups and monoids. The authors connect these algebraic properties to model-theoretic axiomatisability, showing that WRC corresponds to axiomatisability of aleph_0-injective acts, FRE to 2-injective acts, and RIH to a Condition (W) on left acts, thereby bridging semigroup theory with logical frameworks. Collectively, the results map precisely which constructions preserve or destroy WRC/FRE/RIH, yielding practical guidance for constructing coherent semigroups/monoids and clarifying the landscape of axiomatisability for acts.
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.