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''right-ordered group''

A [[Group|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| $  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|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.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.M. Kopytov,   N.Ya. Medvedev,   "The theory of lattice-ordered groups" , Kluwer Acad. Publ.  (1994)  (In Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.T.B. Mura,   A.H. Rhemtulla,   "Orderable groups" , M. Dekker  (1977)</TD></TR></table>