Multiplication tables and word-hyperbolicity in free products of semigroups, monoids, and groups

TL;DR

This paper investigates the preservation of word-hyperbolicity in free products of semigroups, monoids, and groups, demonstrating under mild conditions that such properties are maintained, and extending results to various language-theoretic classes.

Contribution

It proves that free products of word-hyperbolic semigroups, monoids, and groups retain their hyperbolic properties under mild conditions, extending to language-theoretic classes like ET0L.

Findings

Free product of two word-hyperbolic semigroups is word-hyperbolic.

Free product of two word-hyperbolic monoids is word-hyperbolic.

The property extends to free products of groups with ET0L or indexed multiplication tables.

Abstract

This article studies the properties of word-hyperbolic semigroups and monoids, i.e. those having context-free multiplication tables with respect to a regular combing, as defined by Duncan & Gilman. In particular, the preservation of word-hyperbolicity under taking free products is considered. Under mild conditions on the semigroups involved, satisfied e.g. by monoids or regular semigroups, we prove that the semigroup free product of two word-hyperbolic semigroups is again word-hyperbolic. Analogously, with a mild condition on the uniqueness of representation for the identity element, satisfied e.g. by groups, we prove that the monoid free product of two word-hyperbolic monoids is word-hyperbolic. The methods are language-theoretically general, and apply equally well to semigroups, monoids, or groups with a $\mathbf{C}$-multiplication table, where $\mathbf{C}$ is any reversal-closed super-$\operatorname{AFL}$, in the sense of Greibach. In particular, we deduce that the free product of two groups with $\operatorname{ET0L}$ resp. indexed multiplication tables again has an $\operatorname{ET0L}$ resp. indexed multiplication table.