Editing Partially ordered group (section)

{{Short description|Group with a compatible partial order}}
{{redirect|Ordered group|groups with a total or linear order|Linearly ordered group}}
In [[abstract algebra]], a '''partially ordered group''' is a [[group (mathematics)|group]] (''G'', +) equipped with a [[partial order]] "≤" that is ''translation-invariant''; in other words, "≤" has the property that, for all ''a'', ''b'', and ''g'' in ''G'', if ''a'' ≤ ''b'' then ''a'' + ''g'' ≤ ''b'' + ''g'' and ''g'' +'' a'' ≤ ''g'' +'' b''.

An element ''x'' of ''G'' is called '''positive''' if 0 ≤ ''x''. The set of elements 0 ≤ ''x'' is often denoted with ''G''<sup>+</sup>, and is called the '''positive cone of ''G'''''.

By translation invariance, we have ''a'' ≤ ''b'' if and only if 0 ≤ -''a'' + ''b''.
So we can reduce the partial order to a monadic property: {{nobreak|''a'' ≤ ''b''}} [[if and only if]] {{nobreak|-''a'' + ''b'' ∈ ''G''<sup>+</sup>.}}

For the general group ''G'', the existence of a positive cone specifies an order on ''G''. A group ''G'' is a partially orderable group if and only if there exists a subset ''H'' (which is ''G''<sup>+</sup>) of ''G'' such that:
* 0 ∈ ''H''
* if ''a'' ∈ ''H'' and ''b'' ∈ ''H'' then ''a'' + ''b'' ∈ ''H''
* if ''a'' ∈ ''H'' then -''x'' + ''a'' + ''x'' ∈ ''H'' for each ''x'' of ''G''
* if ''a'' ∈ ''H'' and -''a'' ∈ ''H'' then ''a'' = 0

A partially ordered group ''G'' with positive cone ''G''<sup>+</sup> is said to be '''unperforated''' if ''n'' · ''g'' ∈ ''G''<sup>+</sup> for some positive integer ''n'' implies ''g'' ∈ ''G''<sup>+</sup>. Being unperforated means there is no "gap" in the positive cone ''G''<sup>+</sup>.

If the order on the group is a [[linear order]], then it is said to be a [[linearly ordered group]].
If the order on the group is a [[lattice order]], i.e. any two elements have a least upper bound, then it is a '''lattice-ordered group''' (shortly '''l-group''', though usually typeset with a [[Script typeface|script]] l: ℓ-group).

A '''[[Frigyes Riesz|Riesz]] group''' is an unperforated partially ordered group with a property slightly weaker than being a lattice-ordered group. Namely, a Riesz group satisfies the '''Riesz interpolation property''': if ''x''<sub>1</sub>, ''x''<sub>2</sub>, ''y''<sub>1</sub>, ''y''<sub>2</sub> are elements of ''G'' and ''x<sub>i</sub>'' ≤ ''y<sub>j</sub>'', then there exists ''z'' ∈ ''G'' such that ''x<sub>i</sub>'' ≤ ''z'' ≤ ''y<sub>j</sub>''.

If ''G'' and ''H'' are two partially ordered groups, a map from ''G'' to ''H'' is a ''morphism of partially ordered groups'' if it is both a [[group homomorphism]] and a [[monotonic function]]. The partially ordered groups, together with this notion of morphism, form a [[category theory|category]].

Partially ordered groups are used in the definition of [[Valuation (algebra)|valuation]]s of [[field (mathematics)|field]]s.