# Definition:Ordered Group Axioms

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 $\displaystyle a \preceq b \implies \left({c \circ a}\right) \preceq \left({c \circ b}\right)$