Definition:Ordered Group Axioms

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An ordered group is an algebraic structure $\left({G, \circ, \preceq}\right)$ which satisfies the following properties:

\((OG0)\)   $:$   Closure      \(\displaystyle \forall a, b \in G:\) \(\displaystyle a \circ b \in G \)             
\((OG1)\)   $:$   Associativity      \(\displaystyle \forall a, b, c \in G:\) \(\displaystyle a \circ \left({b \circ c}\right) = \left({a \circ b}\right) \circ c \)             
\((OG2)\)   $:$   Identity      \(\displaystyle \exists e \in G: \forall a \in G:\) \(\displaystyle e \circ a = a = a \circ e \)             
\((OG3)\)   $:$   Inverse      \(\displaystyle \forall a \in G: \exists b \in G:\) \(\displaystyle a \circ b = e = b \circ a \)             
\((OG4)\)   $:$   Compatibility of $\preceq$ with $\circ$      \(\displaystyle \forall a, b, c \in G:\) \(\displaystyle a \preceq b \implies \left({a \circ c}\right) \preceq \left({b \circ c}\right) \)             
where $\preceq$ is an ordering    \(\displaystyle a \preceq b \implies \left({c \circ a}\right) \preceq \left({c \circ b}\right) \)             

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