On minimal ideals in pseudo-finite semigroups

TL;DR

This paper investigates the conditions under which right pseudo-finite semigroups possess minimal ideals, showing that in many natural classes such ideals are completely simple, but providing examples where they are absent.

Contribution

It extends previous work by characterizing when minimal ideals exist in right pseudo-finite semigroups across various classes and constructs explicit examples lacking minimal ideals.

Findings

Right pseudo-finiteness often implies a completely simple minimal ideal in certain monoid classes.

Conditions like Green's pre-orders being left compatible with multiplication can ensure the existence of minimal ideals.

Explicit examples of pseudo-finite monoids without minimal ideals are constructed, including regular and $ ext{J}$-trivial cases.

Abstract

A semigroup $S$ is said to be right pseudo-finite if the universal right congruence can be generated by a finite set $U\subseteq S\times S$, and there is a bound on the length of derivations for an arbitrary pair $(s,t)\in S\times S$ as a consequence of those in $U$. This article explores the existence and nature of a minimal ideal in a right pseudo-finite semigroup. Continuing the theme started in an earlier work by Dandan et al., we show that in several natural classes of monoids, right pseudo-finiteness implies the existence of a completely simple minimal ideal. This is the case for orthodox monoids, completely regular monoids and right reversible monoids, which include all commutative monoids. We also show that certain other conditions imply the existence of a minimal ideal, which need not be completely simple; notably, this is the case for semigroups in which one of the Green's pre-orders $\leq_{\mathcal{L}}$ or $\leq_{\mathcal{J}}$ is left compatible with multiplication. Finally, we establish a number of examples of pseudo-finite monoids without a minimal ideal. We develop an explicit construction that yields such examples with additional desired properties, for instance, regularity or $\mathcal{J}$-triviality.