On Subsemigroups and ideals in Free Products of Semigroups

Subsemigroups and ideals of free products of semigroups are studied with respect to the properties of being finitely generated or finitely presented. It is proved that the free product of any two semigroups, at least one of which is nontrivial, contains a two-sided ideal which is not finitely generated as a semigroup, and also contains a subsemigroup which is finitely generated but not finitely presented. By way of contrast, in the free product of two trivial semigroups, every subsemigroup is finitely generated and finitely presented. Further, it is proved that an ideal of a free product of finitely presented semigroups, which is finitely generated as a semigroup, is also finitely presented. It is not known whether one-sided ideals of free products have the same property, but it is shown that they do when the free factors are free commutative.