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A [[Group|group]]  $  G $
on which a [[Partial order|partial order]] relation  $  \leq  $
is given such that for all  $  a , b , x , y $
in  $  G $
the inequality  $  a \leq  b $
implies  $  x a y \leq  x b y $.

The set  $  P = \{ {x \in G } : {x \geq  1 } \} $
in a partially ordered group is called the positive cone, or the integral part, of  $  G $
and satisfies the properties: 1)  $  P P \subseteq P $; 
2)  $  P \cap P  ^ {-} 1 = \{ 1 \} $; 
and 3)  $  x  ^ {-} 1 P x \subseteq P $
for all  $  x \in G $. 
Any subset  $  P $
of  $  G $
that satisfies the conditions 1)–3) induces a partial order on  $  G $(
$  x \leq  y $
if and only if  $  x  ^ {-} 1 y \in P $) 
for which  $  P $
is the positive cone.

Examples of partially ordered groups. The additive group of real numbers with the usual order relation; the group  $  F ( X , \mathbf R ) $
of functions from an arbitrary set  $  X $
into  $  \mathbf R $, 
with the operation

$$ 
( f + g ) ( x)  =  f ( x) + g ( x)
$$

and order relation  $  f \leq  g $
if  $  f ( x) \leq  g( x) $
for all  $  x \in X $; 
the group  $  A ( M) $
of all automorphisms of a totally ordered set  $  M $
with respect to composition of functions, and with order relation  $  \phi \leq  \psi $
if  $  \phi ( m) \leq  \psi ( m) $
for all  $  m \in M $, 
where  $  \phi , \psi \in A ( M) $.

The basic concepts of the theory of partially ordered groups are those of an order homomorphism (cf. [[Ordered group|Ordered group]]), a [[Convex subgroup|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|Totally ordered group]]; [[Lattice-ordered group|Lattice-ordered group]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Birkhoff,   "Lattice theory" , ''Colloq. Publ.'' , '''25''' , Amer. Math. Soc.  (1973)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L. Fuchs,   "Partially ordered algebraic systems" , Pergamon  (1963)</TD></TR></table>