ON THE NUMBER OF SUBSEMIGROUPS OF DIRECT PRODUCTS INVOLVING THE FREE MONOGENIC SEMIGROUP

Abstract

The direct product$\mathbb{N}\times \mathbb{N}$of two free monogenic semigroups contains uncountably many pairwise nonisomorphic subdirect products. Furthermore, the following hold for$\mathbb{N}\times S$, where$S$is a finite semigroup. It contains only countably many pairwise nonisomorphic subsemigroups if and only if$S$is a union of groups. And it contains only countably many pairwise nonisomorphic subdirect products if and only if every element of$S$has a relative left or right identity element.