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:
\(\ds \inf \set {x \circ z, y \circ z}\) | \(=\) | \(\ds \inf \set {x, y} \circ z\) | ||||||||||||
\(\ds \inf \set {z \circ x, z \circ y}\) | \(=\) | \(\ds z \circ \inf \set {x, y}\) |
Proof
First we recall that by definition of ordered group, $\preccurlyeq$ is compatible with $\circ$:
\(\ds \forall x, y, z \in G: \, \) | \(\ds x \preccurlyeq y\) | \(\implies\) | \(\ds x \circ z \preccurlyeq y \circ z\) | |||||||||||
\(\ds \land \ \ \) | \(\ds x \preccurlyeq y\) | \(\implies\) | \(\ds z \circ x \preccurlyeq z \circ y\) |
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:
\(\ds c\) | \(\preccurlyeq\) | \(\ds x\) | Definition of Lower Bound of Set | |||||||||||
\(\, \ds \land \, \) | \(\ds c\) | \(\preccurlyeq\) | \(\ds y\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds c \circ z\) | \(\preccurlyeq\) | \(\ds x \circ z\) | Definition of Relation Compatible with Operation | ||||||||||
\(\, \ds \land \, \) | \(\ds c \circ z\) | \(\preccurlyeq\) | \(\ds y \circ z\) |
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:
\(\ds d' \circ z\) | \(\preccurlyeq\) | \(\ds x \circ z\) | Definition of Lower Bound of Set | |||||||||||
\(\, \ds \land \, \) | \(\ds d' \circ z\) | \(\preccurlyeq\) | \(\ds y \circ z\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds d' \circ z \circ z^{-1}\) | \(\preccurlyeq\) | \(\ds x \circ z \circ z^{-1}\) | Definition of Relation Compatible with Operation | ||||||||||
\(\, \ds \land \, \) | \(\ds d' \circ z \circ z^{-1}\) | \(\preccurlyeq\) | \(\ds y \circ z \circ z^{-1}\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds d'\) | \(\preccurlyeq\) | \(\ds x\) | Group Axiom $\text G 3$: Existence of Inverse Element | ||||||||||
\(\, \ds \land \, \) | \(\ds d'\) | \(\preccurlyeq\) | \(\ds y\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds d'\) | \(\preccurlyeq\) | \(\ds c\) | Definition of Infimum of Set: $d'$ is a lower bound of $\set {x, y}$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds d' \circ z\) | \(\preccurlyeq\) | \(\ds c \circ z\) | Definition of Relation Compatible with Operation |
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 $\set {x \circ z, y \circ z}$.
$\Box$
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:
\(\ds c' \circ z\) | \(\preccurlyeq\) | \(\ds x \circ z\) | Definition of Lower Bound of Set | |||||||||||
\(\, \ds \land \, \) | \(\ds c' \circ z\) | \(\preccurlyeq\) | \(\ds y \circ z\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds c' \circ z \circ z^{-1}\) | \(\preccurlyeq\) | \(\ds x \circ z \circ z^{-1}\) | Definition of Relation Compatible with Operation | ||||||||||
\(\, \ds \land \, \) | \(\ds c' \circ z \circ z^{-1}\) | \(\preccurlyeq\) | \(\ds y \circ z \circ z^{-1}\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds c'\) | \(\preccurlyeq\) | \(\ds x\) | Group Axiom $\text G 3$: Existence of Inverse Element | ||||||||||
\(\, \ds \land \, \) | \(\ds c'\) | \(\preccurlyeq\) | \(\ds y\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds z \circ c'\) | \(\preccurlyeq\) | \(\ds z \circ x\) | Definition of Relation Compatible with Operation | ||||||||||
\(\, \ds \land \, \) | \(\ds z \circ c'\) | \(\preccurlyeq\) | \(\ds z \circ y\) |
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:
\(\ds d\) | \(\preccurlyeq\) | \(\ds z \circ x\) | Definition of Lower Bound of Set | |||||||||||
\(\, \ds \land \, \) | \(\ds d\) | \(\preccurlyeq\) | \(\ds z \circ y\) | |||||||||||
\(\ds z \circ d'\) | \(\preccurlyeq\) | \(\ds z \circ x\) | Definition of $d'$ | |||||||||||
\(\, \ds \land \, \) | \(\ds z \circ d'\) | \(\preccurlyeq\) | \(\ds z \circ y\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds z \circ d'\) | \(\preccurlyeq\) | \(\ds z \circ c'\) | Definition of Infimum of Set: $z \circ d'$ is a lower bound of $\set {z \circ x, z \circ y}$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds d\) | \(\preccurlyeq\) | \(\ds z \circ c'\) | Definition of $d'$ |
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 $\set {z \circ x, z \circ y}$.
$\Box$
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:
\(\ds z \circ c'\) | \(\preccurlyeq\) | \(\ds z \circ x\) | Definition of Lower Bound of Set | |||||||||||
\(\, \ds \land \, \) | \(\ds z \circ c'\) | \(\preccurlyeq\) | \(\ds z \circ y\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds z^{-1} \circ z \circ c'\) | \(\preccurlyeq\) | \(\ds z^{-1} \circ z \circ x\) | Definition of Relation Compatible with Operation | ||||||||||
\(\, \ds \land \, \) | \(\ds z^{-1} \circ z \circ c'\) | \(\preccurlyeq\) | \(\ds z^{-1} \circ z \circ y\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds c'\) | \(\preccurlyeq\) | \(\ds x\) | Group Axiom $\text G 3$: Existence of Inverse Element | ||||||||||
\(\, \ds \land \, \) | \(\ds c'\) | \(\preccurlyeq\) | \(\ds y\) |
Hence $c'$ is a lower bound of $\set {x, y}$.
Let $d$ be a lower bound of $\set {x, y}$.
Then:
\(\ds d\) | \(\preccurlyeq\) | \(\ds x\) | Definition of Lower Bound of Set | |||||||||||
\(\, \ds \land \, \) | \(\ds d\) | \(\preccurlyeq\) | \(\ds y\) | |||||||||||
\(\ds z \circ d\) | \(\preccurlyeq\) | \(\ds z \circ x\) | Definition of $d'$ | |||||||||||
\(\, \ds \land \, \) | \(\ds z \circ d\) | \(\preccurlyeq\) | \(\ds z \circ y\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds z \circ d\) | \(\preccurlyeq\) | \(\ds z \circ c'\) | Definition of Infimum of Set: $z \circ d$ is a lower bound of $\set {z \circ x, z \circ y}$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds z^{-1} \circ z \circ d\) | \(\preccurlyeq\) | \(\ds z^{-1} \circ z \circ c'\) | Definition of Relation Compatible with Operation | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds d\) | \(\preccurlyeq\) | \(\ds c'\) | Definition of $d'$ |
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 $c'$:
- $z \circ \inf \set {x, y} = \inf \set {z \circ x, z \circ y}$
$\Box$
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:
\(\ds \inf \set {x \circ z, y \circ z}\) | \(=\) | \(\ds \inf \set {x, y} \circ z\) | ||||||||||||
\(\ds \inf \set {z \circ x, z \circ y}\) | \(=\) | \(\ds z \circ \inf \set {x, y}\) |
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 15$: Ordered Semigroups: Exercise $15.10 \ \text {(b)}$