"Ordered group" redirects here. For groups with a total or linear order, seeLinearly ordered group.
Inabstract algebra, apartially ordered group is agroup (G, +) equipped with apartial order "≤" that istranslation-invariant; in other words, "≤" has the property that, for alla,b, andg inG, ifa ≤b thena +g ≤b +g andg + a ≤g + b.
An elementx ofG is calledpositive if 0 ≤x. The set of elements 0 ≤x is often denoted withG+, and is called thepositive cone ofG.
By translation invariance, we havea ≤b if and only if 0 ≤ -a +b.So we can reduce the partial order to a monadic property:a ≤bif and only if-a +b ∈G+.
For the general groupG, the existence of a positive cone specifies an order onG. A groupG is a partially orderable group if and only if there exists a subsetH (which isG+) ofG such that:
0 ∈H
ifa ∈H andb ∈H thena +b ∈H
ifa ∈H then -x +a +x ∈H for eachx ofG
ifa ∈H and -a ∈H thena = 0
A partially ordered groupG with positive coneG+ is said to beunperforated ifn ·g ∈G+ for some positive integern impliesg ∈G+. Being unperforated means there is no "gap" in the positive coneG+.
If the order on the group is alinear order, then it is said to be alinearly ordered group.If the order on the group is alattice order, i.e. any two elements have a least upper bound, then it is alattice-ordered group (shortlyl-group, though usually typeset with ascript l: ℓ-group).
ARiesz group is an unperforated partially ordered group with a property slightly weaker than being a lattice-ordered group. Namely, a Riesz group satisfies theRiesz interpolation property: ifx1,x2,y1,y2 are elements ofG andxi ≤yj, then there existsz ∈G such thatxi ≤z ≤yj.
IfG andH are two partially ordered groups, a map fromG toH is amorphism of partially ordered groups if it is both agroup homomorphism and amonotonic function. The partially ordered groups, together with this notion of morphism, form acategory.
Partially ordered groups are used in the definition ofvaluations offields.
A typical example of a partially ordered group isZn, where the group operation is componentwise addition, and we write (a1,...,an) ≤ (b1,...,bn)if and only ifai ≤bi (in the usual order of integers) for alli = 1,...,n.
More generally, ifG is a partially ordered group andX is some set, then the set of all functions fromX toG is again a partially ordered group: all operations are performed componentwise. Furthermore, everysubgroup ofG is a partially ordered group: it inherits the order fromG.
A partially ordered groupG is calledintegrally closed if for all elementsa andb ofG, ifan ≤b for all naturaln thena ≤ 1.[1]
This property is somewhat stronger than the fact that a partially ordered group isArchimedean, though for alattice-ordered group to be integrally closed and to be Archimedean is equivalent.[2]There is a theorem that every integrally closeddirected group is alreadyabelian. This has to do with the fact that a directed group is embeddable into acomplete lattice-ordered group if and only if it is integrally closed.[1]
Elliott, George A. (1976). "On the classification of inductive limits of sequences of semisimple finite-dimensional algebras".Journal of Algebra.38:29–44.doi:10.1016/0021-8693(76)90242-8.
