On the number of subdirect products involving semigroups of integers and natural numbers

TL;DR

This paper generalizes previous results by showing that for various semigroups including integers and natural numbers, the number of subdirect products of their direct products is uncountably infinite, specifically continuum many.

Contribution

It extends known results to a broader class of semigroups, demonstrating that their direct products also have continuum many subdirect products up to isomorphism.

Findings

For each pair of semigroups U, V among Z, N0, N, the product U × V has continuum many subdirect products.

The result generalizes previous findings from positive integers to other semigroups.

There are uncountably many non-isomorphic subdirect products in these semigroup direct products.

Abstract

We extend a recent result that for the (additive) semigroup of positive integers $\mathbb{N}$, there are continuum many subdirect products of $\mathbb{N} \times \mathbb{N}$ up to isomorphism. We prove that for $U,V$ each one of $\mathbb{Z}$ (the group of integers), $\mathbb{N}_{0}$ (the monoid of non-negative integers), or $\mathbb{N}$, we prove that $U \times V$ has continuum many (semigroup) subdirect products up to isomorphism.