# Properties of Ordered Group

## Theorem

Let $\left({G, \circ, \preceq}\right)$ be an ordered group with identity $e$.

Let $x,y,z,w \in G$.

Then the following hold:

### Properties of Ordered Group/OG1

 $(\operatorname{OG}1.1)$ $:$ $\displaystyle x \preceq y \iff x \circ z \preceq y \circ z$ $(\operatorname{OG}1.2)$ $:$ $\displaystyle x \preceq y \iff z \circ x \preceq z \circ y$ $(\operatorname{OG}1.1')$ $:$ $\displaystyle x \prec y \iff x \circ z \prec y \circ z$ $(\operatorname{OG}1.2')$ $:$ $\displaystyle x \prec y \iff z \circ x \prec z \circ y$

#### Corollary: Properties of Ordered Group/OG2

$(\operatorname{OG}2.1):\quad x \preceq y \iff e \preceq y \circ x^{-1}$
$(\operatorname{OG}2.2):\quad x \preceq y \iff e \preceq x^{-1} \circ y$
$(\operatorname{OG}2.3):\quad x \preceq y \iff x \circ y^{-1} \preceq e$
$(\operatorname{OG}2.4):\quad x \preceq y \iff y^{-1} \circ x \preceq e$

$(\operatorname{OG}2.1'):\quad x \prec y \iff e \prec y \circ x^{-1}$
$(\operatorname{OG}2.2'):\quad x \prec y \iff e \prec x^{-1} \circ y$
$(\operatorname{OG}2.3'):\quad x \prec y \iff x \circ y^{-1} \prec e$
$(\operatorname{OG}2.4'):\quad x \prec y \iff y^{-1} \circ x \prec e$

### Inversion Mapping Reverses Ordering in Ordered Group

 $(\text{OG} 3):\quad$ $\displaystyle x \preceq y$ $\iff$ $\displaystyle y^{-1} \preceq x^{-1}$ $(\text{OG}3'):\quad$ $\displaystyle x \prec y$ $\iff$ $\displaystyle y^{-1} \prec x^{-1}$

#### Corollary: Properties of Ordered Group/OG4

 $(\operatorname{OG}4.1)$ $:$ $\displaystyle x \preceq e \iff e \preceq x^{-1}$ $(\operatorname{OG}4.2)$ $:$ $\displaystyle e \preceq x \iff x^{-1} \preceq e$ $(\operatorname{OG}4.1')$ $:$ $\displaystyle x \prec e \iff e \prec x^{-1}$ $(\operatorname{OG}4.2')$ $:$ $\displaystyle e \prec x \iff x^{-1} \prec e$

### Operating on Ordered Group Inequalities

If $x \prec y$ and $z \prec w$, then $x \circ z \prec y \circ w$.

If $x \prec y$ and $z \preceq w$, then $x \circ z \prec y \circ w$.

If $x \preceq y$ and $z \prec w$, then $x \circ z \prec y \circ w$.

If $x \preceq y$ and $z \preceq w$, then $x \circ z \preceq y \circ w$.

### Power Function Strictly Preserves Ordering in Ordered Group

Let $n \in \N_{>0}$ be a strictly positive integer.

If $x \preceq y$ then $x^n \preceq y^n$
If $x \prec y$ then $x^n \prec y^n$

#### Corollary: Properties of Ordered Group/OG5

If $n \in \N_{>0}$ then

$x \preceq e \implies x^n \preceq e$
$e \preceq x \implies e \preceq x^n$
$x \prec e \implies x^n \prec e$
$e \prec x \implies e \prec x^n$