Po-group
partially ordered group
A group 
endowed with a partial order 
such that for all 
,
 |
(Cf. also Partially ordered group.) If 
is the identity of a 
-group 
and 
is the positive cone of 
(cf. 
-group), then the following relations hold:
1) 
;
2) 
;
3) 
for all 
.
If, in a group 
, one can find a set 
with the properties 1)–3), then 
can be made into a 
-group by setting 
if and only if 
. It is correct to identify the order of a 
-group with its positive cone. One often writes a 
-group 
with positive cone 
as 
.
A mapping 
from a 
-group 
into a 
-group 
is an order homomorphism if 
is a homomorphism of the group 
and for all 
,
 |
A homomorphism 
from a 
-group 
into a 
-group 
is an order homomorphism if and only if 
.
A subgroup 
of a 
-group 
is called convex (cf. Convex subgroup) if for all 
with 
,
 |
If 
is a convex subgroup of a 
-group 
, then the set 
of right cosets of 
by 
is a partially ordered set with the induced order 
if there exists an 
such that 
. The quotient group 
of a 
-group 
by a convex normal subgroup 
is a 
-group respect with the induced partial order, and the natural homomorphism 
is an order homomorphism. The homomorphism theorem holds for 
-groups: if 
is an order homomorphism from a 
-group 
into a 
-group 
, then the kernel 
of 
is a convex normal subgroup of 
and there exists an order isomorphism 
from the 
-group 
into 
such that 
.
The most important classes of 
-groups are the class of lattice-ordered groups (cf. 
-group) and the class of totally ordered groups (cf. 
-group).
This article extends and updates the article Partially ordered group (Volume 7).
References
| [a1] | L. Fuchs, "Partially ordered algebraic systems" , Pergamon (1963) |
Po-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Po-group&oldid=48197