Two variants of the Froidure–Pin Algorithm for finite semigroups

Abstract

In this paper, we present two algorithms based on the Froidure-Pin Algorithm for computing the structure of a finite semigroup from a generating set. As was the case with the original algorithm of Froidure and Pin, the algorithms presented here produce the left and right Cayley graphs, a confluent terminating rewriting system, and a reduced word of the rewriting system for every element of the semigroup.

If $U$ is any semigroup, and $A$ is a subset of $U$, then we denote by $⟨A⟩$ the least subsemigroup of $U$ containing $A$. If $B$ is any other subset of $U$, then, roughly speaking, the first algorithm we present describes how to use any information about $⟨A⟩$, that has been found using the Froidure-Pin Algorithm, to compute the semigroup $⟨A\cup B⟩$. More precisely, we describe the data structure for a finite semigroup $S$ given by Froidure and Pin, and how to obtain such a data structure for $⟨A\cup B⟩$ from that for $⟨A⟩$. The second algorithm is a lock-free concurrent version of the Froidure-Pin Algorithm.