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

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

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 $\struct {\Z, +}$.

Let $\struct {\mathscr G, \times_\PP, \supseteq}$ be the ordered structure such that the operation $\times_\PP$ is defined as:

$\forall n \Z, m \Z \in \mathscr G: n \Z \times_\PP m \Z := \paren {n m} \Z$


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

Recall that from Non-Zero Natural Numbers under Multiplication with Divisibility forms Ordered Semigroup, $\struct {\N_{>0}, \times, \divides}$ is indeed an ordered semigroup.

From Natural Numbers with Divisor Operation is Isomorphic to Subgroups of Integer Multiples under Inclusion, we have that $\phi: \struct {\N_{>0}, \divides} \to \struct {\mathscr G, \supseteq}$ is an order isomorphism.

We need to ascertain that $\struct {\mathscr G, \times_\PP, \supseteq}$ is an ordered semigroup.


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Then we need to establish that $\phi: \struct {\N_{>0}, \times} \to \struct {\mathscr G, \times_\PP}$ is a semigroup isomorphism.


This needs considerable tedious hard slog to complete it.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Finish}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.




Sources

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