Definition:Ordered Group

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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.