Natural Numbers with Divisor Operation is Isomorphic to Subgroups of Integer Multiples under Inclusion

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Theorem

Let $\N_{>0}$ denote the set of strictly positive natural numbers.

For $n \in \N_{>0}$, let $n \Z$ denote the set of integer multiples of $n$.

Let $\struct {\Z, +}$ denote the additive group of integers.

Let $\mathscr G$ be the set of all subgroups of $\struct {\Z, +}$.


Consider the algebraic structure $\struct {\N_{>0}, \divides}$, where $\divides$ denotes the divisor operator:

$a \divides b$ denotes that $a$ is a divisor of $b$

Let $\phi: \struct {\N_{>0}, \divides} \to \struct {\mathscr G, \supseteq}$ be the mapping defined as:

$\forall n \in \N_{>0}: \map \phi n = n \Z$

Then $\phi$ is an order isomorphism.


Corollary

Consider the ordered semigroup $\struct {\N_{>0}, \times, \divides}$, where:

$\divides$ denotes the divisor operator:
$a \divides b$ denotes that $a$ is a divisor of $b$
$\times$ denotes integer multiplication.

Let $\phi: \struct {\N_{>0}, \times, \divides} \to \struct {\mathscr G, \times_\PP, \supseteq}$ be the mapping defined as:

$\forall n \in \N_{>0}: \map \phi n = n \Z$

Then $\phi$ is an ordered semigroup isomorphism.


Proof

We note that from Subgroups of Additive Group of Integers, the subgroups of $\struct {\Z, +}$ are precisely the sets of integer multiples $n \Z$, for $n \in \N_{>0}$.

For each $n \in \N_{>0}$, there is a unique $n \Z \in \mathscr G$.

Hence $\phi$ is a bijection.


It remains to be demonstrated that $\phi$ is order-preserving in both directions.


Thus:

\(\ds a\) \(\divides\) \(\ds b\)
\(\ds \leadstoandfrom \ \ \) \(\ds \exists k \in \Z: \, \) \(\ds k a\) \(=\) \(\ds b\) Definition of Divisor of Integer


Then:

\(\ds x\) \(\in\) \(\ds b \Z\)
\(\ds \leadstoandfrom \ \ \) \(\ds \exists m \in \Z: \, \) \(\ds x\) \(=\) \(\ds m b\) Definition of Set of Integer Multiples
\(\ds \leadstoandfrom \ \ \) \(\ds \exists k \in \Z: \, \) \(\ds x\) \(=\) \(\ds \paren {m k} a\) a priori: $k a = b$
\(\ds \leadstoandfrom \ \ \) \(\ds x\) \(\in\) \(\ds a \Z\) Definition of Set of Integer Multiples
\(\ds \leadstoandfrom \ \ \) \(\ds a \Z\) \(\supseteq\) \(\ds b \Z\) Definition of Subset

Hence the result.

$\blacksquare$


Sources

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