Two variants of the Froiduire-Pin Algorithm for finite semigroups

TL;DR

This paper introduces two new algorithms based on the Froidure-Pin Algorithm for computing finite semigroup structures, including Cayley graphs and rewriting systems, with one algorithm enhancing incremental computation and the other enabling lock-free concurrency.

Contribution

It presents a method to extend existing semigroup data structures incrementally and introduces a lock-free concurrent algorithm for improved computational efficiency.

Findings

Efficient computation of semigroup structures from generating sets.

Incremental algorithm for combining semigroups using existing data.

Lock-free concurrent algorithm for semigroup computation.

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 $\langle A\rangle$ 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 $\langle A\rangle$, that has been found using the Froidure-Pin Algorithm, to compute the semigroup $\langle A\cup B\rangle$. 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 $\langle A\cup B\rangle$ from that for $\langle A\rangle$. The second algorithm is a lock-free concurrent version of the Froidure-Pin Algorithm.