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

In this paper, we extend a recent result that for the (additive) semigroup of positive integers [Formula: see text], there are continuum many subdirect products of [Formula: see text] up to isomorphism. We prove that for [Formula: see text] each one of [Formula: see text] (the group of integers), [Formula: see text] (the monoid of non-negative integers), or [Formula: see text], the direct product [Formula: see text] contains continuum many (semigroup) subdirect products up to isomorphism.