Malcev presentations for subsemigroups of groups — a survey

Summary

Abstract

This paper introduces and surveys the theory of Malcev presentations. [Malcev presentations are a species of presentation that can be used to define any semigroup that can be embedded into a group.] In particular, various classes of groups and monoids all of whose finitely generated subsemigroups admit finite Malcev presentations are described; closure and containment results are stated; links with the theory of automatic semigroups are mentioned; and various questions asked. Many of the results stated herein are summarized in tabular form.

Introduction

A Malcev presentation is a presentation of a special type for a semigroup that embeds in a group. Informally, a Malcev presentation defines a semigroup by means of generators, defining relations, and the unwritten rule that the semigroup so defined must be embeddable in a group. This rule of group-embeddability is worth an infinite number of defining relations, in the sense that a semigroup may admit a finite Malcev presentation but no finite ‘ordinary’ presentation. Spehner [31] introduced Malcev presentations, though they are based on Malcev's necessary and sufficient condition for a semigroup to be embeddable into a group [22]. Spehner exhibited an example of a finitely generated submonoid of a free monoid that admitted a finite Malcev presentation but which was not finitely presented. He later showed that all finitely generated submonoids of free monoids have finite Malcev presentations [32]. Until the recent work of Cain, Robertson & Ruškuc [12, 11, 10, 9, 8], Spehner's two articles represented the whole of the literature on Malcev presentations.