Infima in Ordered Group

Theorem
Let $\struct {G, \circ, \preccurlyeq}$ be an ordered group.

Let $x, y, z \in G$ be arbitrary.

Let any one of the sets $\set {x, y}$, $\set {x \circ z, y \circ z}$ or $\set {z \circ x, z \circ y}$ admit an infimum.

Then all three sets admit an infimum, and:

Proof
First we recall that by definition of ordered group, $\preccurlyeq$ is compatible with $\circ$:

Let $\set {x, y}$ admit an infimum $c$.

Then by definition of infimum:


 * $(1): \quad c$ is a lower bound of $\set {x, y}$ in $G$
 * $(2): \quad d \preccurlyeq c$ for all lower bounds $d$ of $\set {x, y}$ in $S$.

Thus we have:

Hence $\inf \set {x, y} \circ z$ is a lower bound of $\set {x \circ z, y \circ z}$.

Let $d$ be a lower bound of $\set {x \circ z, y \circ z}$.

Then as $G$ is a group we have that:
 * $d = d' \circ z$

for some $d' \in G$.

Then:

Hence $\inf \set {x, y} \circ z$ is a lower bound of $\set {x \circ z, y \circ z}$ which is smaller than an arbitrary lower bound $d$ of $\set {x \circ z, y \circ z}$.

That is, $\inf \set {x, y} \circ z$ is an infimum of $\set {x \circ z, y \circ z}$.

Let $\set {x \circ z, y \circ z}$ admit an infimum $c$.

Then by definition of infimum:


 * $(1): \quad c$ is a lower bound of $\set {x \circ z, y \circ z}$ in $G$
 * $(2): \quad d \preccurlyeq c$ for all lower bounds $d$ of $\set {x \circ z, y \circ z}$ in $S$.

As $G$ is a group, there exists $c' \in G$ such that $c' \circ z = c$.

Thus we have:

Hence $z \circ c'$ is a lower bound of $\set {z \circ x, z \circ y}$.

Let $d$ be a lower bound of $\set {z \circ x, z \circ y}$.

As $G$ is a group, there exists $d' \in G$ such that $z \circ d' = d$.

Then:

Hence $z \circ c'$ is a lower bound of $\set {z \circ x, z \circ y}$ which is smaller than an arbitrary lower bound $d$ of $\set {z \circ x, z \circ y}$.

That is, $z \circ c'$ is an infimum of $\set {z \circ x, z \circ y}$.

Let $\set {z \circ x, z \circ y}$ admit an infimum $c$.

Then by definition of infimum:


 * $(1): \quad c$ is a lower bound of $\set {z \circ x, z \circ y}$ in $G$
 * $(2): \quad d \preccurlyeq c$ for all lower bounds $d$ of $\set {z \circ x, z \circ y}$ in $S$.

As $G$ is a group, there exists $c' \in G$ such that $z \circ c' = c$.

Thus we have:

Hence $c'$ is a lower bound of $\set {x, y}$.

Let $d$ be a lower bound of $\set {x, y}$.

Then:

Hence $c'$ is a lower bound of $\set {x, y}$ which is smaller than an arbitrary lower bound $d$ of $\set {x, y}$.

That is, $c'$ is an infimum of $\set {x, y}$.

Hence by definition of $c'$:
 * $z \circ \inf \set {x, y} = \inf \set {z \circ x, z \circ y}$

Thus we have shown that if any of the three sets $\set {x, y}$, $\set {x \circ z, y \circ z}$ or $\set {z \circ x, z \circ y}$ admit an infimum, they all do, and: