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 non-isomorphic subdirect products. Furthermore, the following hold for $\mathbb{N}\times S$, where $S$ is a finite semigroup. It contains only countably many pairwise non-isomorphic subsemigroups if and only if $S$ is a union of groups. And it contains only countably many pairwise non-isomorphic subdirect products if and only if every element of $S$ has a relative left- or right identity element.