Order Embedding/Examples/Finite Subsets of Natural Numbers in Divisibility Structure

Example of Order Embedding

Consider the relational structures:

$\struct {\Z_{>0}, \divides}$, where $\Z_{>0}$ denotes the strictly positive integers and $\divides$ denotes the divisor relation
$\struct {\FF, \subseteq}$, where $\FF$ denotes the finite subsets of the natural numbers without zero $\N_{\ne 0}$ and $\subseteq$ denotes the subset relation.

Let $\pi: \FF \to \Z_{>0}$ be the mapping defined as:

$\forall S \in \FF: \map \pi S = \ds \prod_{n \mathop \in S} \map p n$

where $\map p n$ denotes the $n$th prime number:

$\map p 1 = 2, \map p 2 = 3, \map p 3 = 5, \ldots$

Then $\pi$ is an order embedding of $\FF$ into $\Z_{>0}$.

Proof

Let $S \subseteq T$ where $S, T \in \F$.

Then:

 $\ds S$ $\subseteq$ $\ds T$ $\ds \leadstoandfrom \ \$ $\ds \forall n \in S: \,$ $\ds n$ $\in$ $\ds T$ $\ds \leadstoandfrom \ \$ $\ds \prod_{n \mathop \in S} \map p n$ $\divides$ $\ds \prod_{n \mathop \in T} \map p n$ $\ds \leadstoandfrom \ \$ $\ds \map \pi S$ $\divides$ $\ds \map \pi T$

$\blacksquare$