Editing Partially ordered group (section)

== Examples ==

* The [[integer]]s with their usual order
* An [[ordered vector space]] is a partially ordered group
* A [[Riesz space]] is a lattice-ordered group
* A typical example of a partially ordered group is '''[[integer|Z]]'''<sup>''n''</sup>, where the group operation is componentwise addition, and we write (''a''<sub>1</sub>,...,''a''<sub>''n''</sub>) ≤ (''b''<sub>1</sub>,...,''b''<sub>''n''</sub>) [[if and only if]] ''a''<sub>''i''</sub> ≤ ''b''<sub>''i''</sub> (in the usual order of integers) for all ''i'' = 1,..., ''n''.
* More generally, if ''G'' is a partially ordered group and ''X'' is some set, then the set of all functions from ''X'' to ''G'' is again a partially ordered group: all operations are performed componentwise. Furthermore, every [[subgroup]] of ''G'' is a partially ordered group: it inherits the order from ''G''.
* If ''A'' is an [[approximately finite-dimensional C*-algebra]], or more generally, if ''A'' is a stably finite unital C*-algebra, then [[Approximately finite-dimensional C*-algebra#K0|K<sub>0</sub>]](''A'') is a partially ordered [[abelian group]]. (Elliott, 1976)