On the word problem for free products of semigroups and monoids

Abstract

We study the language-theoretic aspects of the word problem, in the sense of Duncan & Gilman, of free products of semigroups and monoids. First, we provide algebraic tools for studying classes of languages known as super-AFLs, which generalise e.g. the context-free or the indexed languages. When C is a super-AFL closed under reversal, we prove that the semigroup (monoid) free product of two semigroups (resp. monoids) with word problem in C also has word problem in C. This recovers and generalises a recent result by Brough, Cain & Pfeiffer that the class of context-free semigroups (monoids) is closed under taking free products. As a group-theoretic corollary, we deduce that the word problem of the (group) free product of two groups with word problem in C is also in C. As a particular case, we find that the free product of two groups with indexed word problem has indexed word problem.