Po-group

partially ordered group

A group $ \{ G; \cdot, \cle \} $ endowed with a partial order $ \cle $ such that for all $ x,y,z,t \in G $,

$$ x \cle y \Rightarrow zxt \cle xyt. $$

(Cf. also Partially ordered group.) If $ e $ is the identity of a $ po $- group $ G $ and $ P = P ( G ) = \{ {x \in G } : {x \cge e } \} $ is the positive cone of $ G $( cf. $ l $- group), then the following relations hold:

1) $ P \cdot P \subseteq P $;

2) $ P \cap P = \{ e \} $;

3) $ x ^ {- 1 } Px \subseteq P $ for all $ x $.

If, in a group $ G $, one can find a set $ P $ with the properties 1)–3), then $ G $ can be made into a $ po $- group by setting $ x \cle y $ if and only if $ yx ^ {- 1 } \in P $. It is correct to identify the order of a $ po $- group with its positive cone. One often writes a $ po $- group $ G $ with positive cone $ P $ as $ ( G,P ) $.

A mapping $ \varphi : G \rightarrow H $ from a $ po $- group $ G $ into a $ po $- group $ H $ is an order homomorphism if $ \varphi $ is a homomorphism of the group $ G $ and for all $ x,y \in G $,

$$ x \cle y \Rightarrow \varphi ( x ) \cle \varphi ( y ) . $$

A homomorphism $ \varphi $ from a $ po $- group $ ( G,P ) $ into a $ po $- group $ ( H,Q ) $ is an order homomorphism if and only if $ \varphi ( P ) \subseteq Q $.

A subgroup $ H $ of a $ po $- group $ G $ is called convex (cf. Convex subgroup) if for all $ x,y,z $ with $ x,z \in H $,

$$ x \cle y \cle z \Rightarrow y \in H. $$

If $ H $ is a convex subgroup of a $ po $- group $ G $, then the set $ G/H $ of right cosets of $ G $ by $ H $ is a partially ordered set with the induced order $ Hx \cle Hy $ if there exists an $ h \in H $ such that $ x \cle hy $. The quotient group $ G/H $ of a $ po $- group $ G $ by a convex normal subgroup $ H $ is a $ po $- group respect with the induced partial order, and the natural homomorphism $ \tau : G \rightarrow {G/H } $ is an order homomorphism. The homomorphism theorem holds for $ po $- groups: if $ \varphi $ is an order homomorphism from a $ po $- group $ G $ into a $ po $- group $ H $, then the kernel $ N = \{ {x \in G } : {\varphi ( x ) = e } \} $ of $ \varphi $ is a convex normal subgroup of $ G $ and there exists an order isomorphism $ \psi $ from the $ po $- group $ G/N $ into $ H $ such that $ \varphi = \tau \psi $.

The most important classes of $ po $- groups are the class of lattice-ordered groups (cf. $ l $- group) and the class of totally ordered groups (cf. $ o $- group).

This article extends and updates the article Partially ordered group (Volume 7).

References

[a1]	L. Fuchs, "Partially ordered algebraic systems" , Pergamon (1963)