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

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.