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

TL;DR

The paper proves that for any pair $(U,V)$ with $U,V\in\{\mathbb{Z},\mathbb{N}_0,\mathbb{N}\}$, the direct product $U\times V$ contains continuum many pairwise non-isomorphic semigroup subdirect products. It achieves this by constructing a family $S_\sigma$ of subsemigroups of $\mathbb{Z}\times\mathbb{Z}$ indexed by sequences $\sigma=(c_i)_{i\ge2}$ with $c_2=1$ and $c_{i+1}\ge 2c_i$, and showing that the intersections $S_\sigma\cap(U\times V)$ are subdirect products whose isomorphism type is dictated by $\sigma$. A key technical tool is a precise description of indecomposable elements on each intersected semigroup, which remains distinct for different $\sigma$, ensuring uncountably many non-isomorphic subdirect products across all $(U,V)$ in the listed set. The results reveal rich, non-fiber-product subdirect-product behavior in semigroups and contrast with the limited subdirect-product structure in the corresponding group case; they also raise open questions about when such abundance can fail in other infinite semigroups.

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.

Figures (1)

• Figure 1: The semigroup $S_\sigma$, with $\sigma=(1,2,5,10,\dots)$.

Theorems & Definitions (52)

• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Proposition 3.3
• proof