On the number of subsemigroups of direct products involving the free monogenic semigroup

TL;DR

This paper investigates the structure and quantity of subsemigroups and subdirect products in direct products involving the free monogenic semigroup, revealing conditions for countability and uncountability.

Contribution

It characterizes when direct products with finite semigroups have countably or uncountably many non-isomorphic subsemigroups or subdirect products.

Findings

$ ewline ext{The direct product } ewline ext{of two free monogenic semigroups has uncountably many non-isomorphic subsemigroups.} ewline ext{For } ewline ext{$ ewline ext{$ ext{N} imes S$ with finite } S, ext{ the countability depends on } S ext{ being a union of groups or having relative identities.} ewline ext{Conditions for countability are characterized precisely.}

The structure of subsemigroups in these products is deeply linked to algebraic properties of the finite semigroup } S.

The results provide a clear dichotomy based on the properties of } S ext{, distinguishing between countable and uncountable cases.

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.