Parallel algorithms for computing finite semigroups

Abstract

In this paper, we present two algorithms based on the Froidure-Pin Algorithm for computing a finite semigroup. If U is any semigroup, and A be 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, 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, B⟩ from that for ⟨A⟩. The second algorithm is a lock-free concurrent version of the Froidure-Pin Algorithm. 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 they output.