# Ro-group

right-ordered group

A group $G$ endowed with a total order $\le$ such that for all $x,y,z \in G$,

$$x \le y \Rightarrow xz \le yz.$$

If $P = P ( G ) = \{ {x \in G } : {x \ge e } \}$ is the positive cone of the $ro$- group $G$( cf. also $l$- group), then:

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

2) $P \cap P ^ {- 1 } = \{ e \}$;

3) $P \cup P ^ {- 1 } = G$. If, in a group $G$, there is a subset $P$ satisfying 1)–3), then $G$ can given the structure of a $ro$- group with positive cone $P$ by a setting $x \le y$ if and only if $yx ^ {- 1 } \in P$. The positive cone of a $ro$- group is isolated, i.e., $x ^ {n} \in P \Rightarrow x \in P$.

The group of order automorphisms ${ \mathop{\rm Aut} } ( X )$ of a totally ordered set $\{ X; \le \}$ can be turned into a $ro$- group by defining the following relation $\le$ on it. Let $\prec$ be any well ordering on $X$: $x _ {1} \prec \dots \prec x _ \alpha \prec \dots$. Let $\varphi \in { \mathop{\rm Aut} } ( X )$ and let $x _ \alpha$ be the first (with respect to $\prec$) element in $\{ {x \in X } : {x \varphi \neq x } \}$. Then $A ( X )$ is a $ro$- group with respect to the order with positive cone

$$P \subset A ( X ) = \left \{ {\varphi \in { \mathop{\rm Aut} } ( X ) } : {x _ \alpha \varphi \ge x _ \alpha } \right \} .$$

Any $ro$- group is isomorphic to a subgroup of the $ro$- group ${ \mathop{\rm Aut} } ( X )$ for some totally ordered set $X$. There exist simple $ro$- groups whose finitely generated subgroups coincide with the commutator subgroup. The class of all groups that can be turned into a $ro$- group is a quasi-variety, i.e., it is defined by a system of formulas of the form:

$$\forall x _ {1} \dots x _ {n} :$$

$$( w _ {1} ( x _ {1} \dots x _ {n} ) = e \& \dots \& w _ {m} ( x _ {1} \dots x _ {n} ) = e ) \Rightarrow$$

$$\Rightarrow w ( x _ {1} \dots x _ {n} ) = e,$$

where $w$, $w _ {i}$ are the group-theoretical words. This class is closed under formation of subgroups, Cartesian and free products, and extension, and is locally closed.

The system ${\mathcal C} ( G )$ of convex subgroups of a $ro$- group $G$ is a complete chain. It can be non-solvable, non-infra-invariant and non-normal. There exist non-Abelian $ro$- groups without proper convex subgroups.

A $ro$- group $G$ is Archimedean if for any positive elements $x,y \in G$ there exists a positive integer $n$ such that $x ^ {n} > y$. An Archimedean $ro$- group is order-isomorphic to some subgroup of the additive group $\mathbf R$ of real numbers with the natural order. The class of Conradian $ro$- groups, i.e., $ro$- groups for which the system ${\mathcal C} ( G )$ is subnormal and the quotient groups of the jumps of ${\mathcal C} ( G )$ are Archimedean, is well investigated.

How to Cite This Entry:
Ro-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ro-group&oldid=53509
This article was adapted from an original article by V.M. Kopytov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article