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)