partially ordered group
A partially ordered group is a group that is a poset at the same time, such that if and , then
for any . The two conditions are equivalent to the one condition for all . A partially ordered group is also called a po-group for short.
One of the immediate properties of a po-group is this: if , then . To see this, left multiply by the first inequality by on both sides to obtain . Then right multiply the resulting inequality on both sides by to obtain the desired inequality: .
A po-group whose underlying poset is a lattice is called a lattice ordered group, or an l-group.
Since the definition above does not involve any specific group axioms, one can more generally introduce partial ordering on a semigroup in the same fashion. The result is called a partially ordered semigroup, or a po-semigroup for short. A lattice ordered semigroup is defined similarly.
|Title||partially ordered group|
|Date of creation||2013-03-22 16:42:25|
|Last modified on||2013-03-22 16:42:25|
|Last modified by||CWoo (3771)|
|Synonym||integrally closed po-group|
|Defines||lattice ordered group|
|Defines||Archimedean partially ordered group|
|Defines||integrally closed group|
|Defines||integrally closed partially ordered group|
|Defines||partially ordered semigroup|
|Defines||lattice ordered semigroup|