# Definition:Ordered Group

## Definition

An ordered group is an ordered structure $\struct {G, \circ, \preceq}$ such that $\struct {G, \circ}$ is a group.

### Ordered Group Axioms

The properties that define an ordered group can be gathered together as follows:

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 $\ds a \preceq b \implies \paren {c \circ a} \preceq \paren {c \circ b}$

## Also see

• Results about ordered groups can be found here.