Subsets Greater Than and Less Than Identity of Ordered Abelian Group are Isomorphic Ordered Semigroups

Theorem
Let $\struct {G, \circ, \preccurlyeq}$ be an ordered abelian group whose identity element is $e$.

Let $G^+$ and $G^-$ denote the subsets of $G$ defined as:
 * $G^+ = \set {x \in G: e \preccurlyeq x}$
 * $G^- = \set {x \in G: x \preccurlyeq e}$

Then $\struct {G^+, \circ, \preccurlyeq}$ and $\struct {G^-, \circ, \succcurlyeq}$ are subsemigroups of $G$ such that the inversion mapping $\iota: G \to G$ defined as:
 * $\forall g \in G: \map \iota g = g^{-1}$

is:
 * an isomorphism from $\struct {G^+, \circ, \preccurlyeq}$ to $\struct {G^-, \circ, \succcurlyeq}$

and:
 * an isomorphism from $\struct {G^-, \circ, \preccurlyeq}$ to $\struct {G^+, \circ, \succcurlyeq}$

Proof
Let $x, y \in G^+$ be arbitrary.

Then:

So by the Subsemigroup Closure Test $\struct {G^+, \circ}$ is a subsemigroup of $\struct {G, \circ}$.

Let $x, y \in G^-$ be arbitrary.

Then:

So by the Subsemigroup Closure Test $\struct {G^-, \circ}$ is a subsemigroup of $\struct {G, \circ}$.

Then we establish that:

demonstrating that for all $\iota$ has the morphism property over the whole of $G$.

Then we note that:

Similarly:

Then we note that:

demonstrating that $\iota$ is an injection.

Let $x \in G^+$.

Then :
 * $\exists x^{-1} \in G^-: \map \iota {x^{-1} } = x$

Similarly, let $x \in G^-$.

Then :
 * $\exists x^{-1} \in G^+: \map \iota {x^{-1} } = x$

So $\iota: G^+ \to G^-$ and $\iota: G^- \to G^+$ are both surjections.

Hence $\iota: G^+ \to G^-$ and $\iota: G^- \to G^+$ are both bijections by definition.

Thus we have that:
 * $\iota: \struct {G^+, \circ} \to \struct {G^-, \circ}$ is a semigroup isomorphism

and
 * $\iota: \struct {G^-, \circ} \to \struct {G^+, \circ}$ is a semigroup isomorphism.

Let $x, y \in G^+$ be arbitrary, such that $x \preccurlyeq y$.

We have:

Let $x, y \in G^-$ be arbitrary, such that $x \succcurlyeq y$.

We have:

Thus we have demonstrated that $\iota: \struct {G^+, \preccurlyeq} \to \struct {G^-, \succcurlyeq}$ and $\iota: \struct {G^-, \succcurlyeq} \to \struct {G^+, \preccurlyeq}$ are order isomorphisms.

So we have that:

Thus we have that:
 * $\iota: \struct {G^+, \circ} \to \struct {G^-, \circ}$ is a semigroup isomorphism
 * $\iota: \struct {G^+, \preccurlyeq} \to \struct {G^-, \succcurlyeq}$ is an order isomorphism

and so:
 * $\iota: \struct {G^+, \circ, \preccurlyeq} \to \struct {G^-, \circ, \succcurlyeq}$ is an ordered semigroup isomorphism

and
 * $\iota: \struct {G^-, \circ} \to \struct {G^+, \circ}$ is a semigroup isomorphism
 * $\iota: \struct {G^-, \succcurlyeq} \to \struct {G^+, \preccurlyeq}$ is an order isomorphism

and so:
 * $\iota: \struct {G^-, \circ, \succcurlyeq} \to \struct {G^+, \circ, \preccurlyeq}$ is an ordered semigroup isomorphism