Partially ordered group

A group $ G $ on which a 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), 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)