On semigroup presentations

TLDR

Semigroup presentations have long been studied as examples of semigroups, and in 1967 Neumann introduced an enumeration method analogous to Todd–Coxeter, later proved by Jura in 1978. The paper aims to implement Neumann’s enumeration algorithm on a computer and to analyze specific semigroup presentations whose related group presentations have produced interesting groups. The authors implement Neumann’s enumeration algorithm on a computer and use the resulting program to guide algebraic proofs of new theorems about semigroup presentations. The computer‑guided enumeration led to new algebraic theorems about semigroup presentations.

Abstract

Semigroup presentations have been studied over a long period, usually as a means of providing examples of semigroups. In 1967 B. H. Neumann introduced an enumeration method for finitely presented semigroups analogous to the Todd–Coxeter coset enumeration process for groups. A proof of Neumann's enumeration method was given by Jura in 1978. In Section 3 of this paper we describe a machine implementation of a semigroup enumeration algorithm based on that of Neumann. In Section 2 we examine certain semigroup presentations, motivated by the fact that the corresponding group presentation has yielded interesting groups. The theorems, although proved algebraically, were suggested by the semigroup enumeration program.