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arXiv · 2112.10665

On the Word Problem for Free Products of Semigroups and Monoids

Carl-Fredrik Nyberg-Brodda
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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.

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Carl-Fredrik Nyberg-Brodda. 2021-12-20. On the Word Problem for Free Products of Semigroups and Monoids. https://arxiv.org/abs/2112.10665

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