On the Word Problem for Free Products of Semigroups and Monoids

TL;DR

This paper investigates the language-theoretic properties of the word problem in free products of semigroups and monoids, extending known results to broader classes of languages called super-AFLs.

Contribution

It introduces algebraic tools for super-AFLs and proves closure of the word problem under free products for these classes, generalizing previous results for context-free languages.

Findings

Closure of super-AFLs under free products for word problems

Extension of context-free closure results to super-AFLs

Application to group word problems, including indexed languages

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 $\mathcal{C}$ is a super-AFL closed under reversal, we prove that the semigroup (monoid) free product of two semigroups (resp. monoids) with word problem in $\mathcal{C}$ also has word problem in $\mathcal{C}$. 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 $\mathcal{C}$ is also in $\mathcal{C}$. As a particular case, we find that the free product of two groups with indexed word problem has indexed word problem.