Small overlap monoids I: The word problem

Abstract

We develop a combinatorial approach to the study of semigroups and monoids with finite presentations satisfying small overlap conditions. In contrast to existing geometric methods, our approach facilitates a sequential left–right analysis of words which lends itself to the development of practical, efficient computational algorithms. In particular, we obtain a highly practical linear time solution to the word problem for monoids and semigroups with finite presentations satisfying the condition C ( 4 ) , and a polynomial time solution to the uniform word problem for presentations satisfying the same condition.