On Residual Finiteness of Monoids, their Schutzenberger Groups and Associated Actions

TL;DR

This paper explores the relationships between residual finiteness properties of monoids, their Schutzenberger groups, and associated actions, providing comprehensive answers across various classes of monoids.

Contribution

It systematically analyzes when residual finiteness of monoids implies or is implied by residual finiteness of related structures, across multiple monoid classes.

Findings

Complete characterization for arbitrary monoids.

Results for regular monoids and those with finitely many classes.

Conditions under which residual finiteness properties are equivalent.

Abstract

In this paper we discuss connections between the following properties: (RFM) residual finiteness of a monoid M; (RFSG) residual finiteness of Schutzenberger groups of M; and (RFRL) residual finiteness of the natural actions of M on its Green's R- and L-classes. The general question is whether (RFM) implies (RFSG) and/or (RFRL), and vice versa. We consider these questions in all the possible combinations of the following situations: M is an arbitrary monoid; M is an arbitrary regular monoid; every J-class of M has finitely many R- and L-classes; M has finitely many left and right ideals. In each case we obtain complete answers.