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 $langle Arangle$ 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 $langle Arangle$, that has been found using the Froidure-Pin Algorithm, to compute the semigroup $langle Acup Brangle$. 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 $langle Acup Brangle$ from that for $langle Arangle$. The second algorithm is a lock-free concurrent version of the Froidure-Pin Algorithm.
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