Partially ordered group
A group on which a partial order relation
is given such that for all
in
the inequality
implies
.
The set in a partially ordered group is called the positive cone, or the integral part, of
and satisfies the properties: 1)
; 2)
; and 3)
for all
. Any subset
of
that satisfies the conditions 1)–3) induces a partial order on
(
if and only if
) for which
is the positive cone.
Examples of partially ordered groups. The additive group of real numbers with the usual order relation; the group of functions from an arbitrary set
into
, with the operation
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and order relation if
for all
; the group
of all automorphisms of a totally ordered set
with respect to composition of functions, and with order relation
if
for all
, where
.
The basic concepts of the theory of partially ordered groups are those of an order homomorphism (cf. Ordered group), a convex subgroup, and Cartesian and lexicographic products.
Important classes of partially ordered groups are totally ordered groups and lattice-ordered groups (cf. Totally ordered group; Lattice-ordered group).
References
[1] | G. Birkhoff, "Lattice theory" , Colloq. Publ. , 25 , Amer. Math. Soc. (1973) |
[2] | L. Fuchs, "Partially ordered algebraic systems" , Pergamon (1963) |
Partially ordered group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Partially_ordered_group&oldid=48137