Axiom:Ordered Group Axioms

Definition

An ordered group is an algebraic system $\struct {G, \circ, \preceq}$ which satisfies the following properties:

$(\text {OG} 0)$	$:$	Closure	$\ds \forall a, b \in G:$	$\ds a \circ b \in G $
$(\text {OG} 1)$	$:$	Associativity	$\ds \forall a, b, c \in G:$	$\ds a \circ \paren {b \circ c} = \paren {a \circ b} \circ c $
$(\text {OG} 2)$	$:$	Identity	$\ds \exists e \in G: \forall a \in G:$	$\ds e \circ a = a = a \circ e $
$(\text {OG} 3)$	$:$	Inverse	$\ds \forall a \in G: \exists b \in G:$	$\ds a \circ b = e = b \circ a $
$(\text {OG} 4)$	$:$	Compatibility of $\preceq$ with $\circ$	$\ds \forall a, b, c \in G:$	$\ds a \preceq b \implies \paren {a \circ c} \preceq \paren {b \circ c} $
		where $\preceq$ is an ordering		$\ds a \preceq b \implies \paren {c \circ a} \preceq \paren {c \circ b} $

These stipulations can be referred to as the ordered group axioms.

\((\text {OG} 0)\)	$:$	Closure	\(\ds \forall a, b \in G:\)	\(\ds a \circ b \in G \)
\((\text {OG} 1)\)	$:$	Associativity	\(\ds \forall a, b, c \in G:\)	\(\ds a \circ \paren {b \circ c} = \paren {a \circ b} \circ c \)
\((\text {OG} 2)\)	$:$	Identity	\(\ds \exists e \in G: \forall a \in G:\)	\(\ds e \circ a = a = a \circ e \)
\((\text {OG} 3)\)	$:$	Inverse	\(\ds \forall a \in G: \exists b \in G:\)	\(\ds a \circ b = e = b \circ a \)
\((\text {OG} 4)\)	$:$	Compatibility of $\preceq$ with $\circ$	\(\ds \forall a, b, c \in G:\)	\(\ds a \preceq b \implies \paren {a \circ c} \preceq \paren {b \circ c} \)
		where $\preceq$ is an ordering		\(\ds a \preceq b \implies \paren {c \circ a} \preceq \paren {c \circ b} \)