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
 |
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