  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.
