Difference between revisions of "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).