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Half-ordered groups. (Groupes à moitié ordonnés.) (French) Zbl 0766.06014
Let $$G$$ be a group. Suppose that $$\leq$$ is a partial order on $$G$$. The set of all $$x\in G$$ such that for each $$y,z\in G$$ the implication $$y\leq z\Rightarrow xy\leq xz$$ holds will be denoted by $$G \uparrow$$. The set $$G \downarrow$$ is defined analogously by applying the implication $$y\leq z\Rightarrow xy\geq xz$$. The authors call $$G$$ a half-ordered group if the following conditions are satisfied: 1) the partial order $$\leq$$ is non- trivial; 2) $$y\leq z\Rightarrow yx\leq zx$$ for each $$x,y,z\in G$$; 3) $$G=G \uparrow\cup G \downarrow$$. If, moreover, $$G \uparrow$$ is linearly ordered (or lattice-ordered), then $$G$$ is said to be a half linearly ordered group (or a half lattice-ordered group, respectively).
For a chain $$T$$ let $$M(T)$$ be the group of all monotone permutations of $$T$$; let $$M(T)$$ be partially ordered under the pointwise order. Then $$M(T)$$ is a half-ordered group.
The main results of the paper concern the case when $$G \uparrow$$ is linearly ordered or lattice-ordered. It is proved that if $$G \uparrow$$ is linearly ordered, then (i) $$G \downarrow$$ is the set $$\{x\in G: x\neq e$$ and $$x^ 2=e\}$$; (ii) if $$x\in G \downarrow$$ and $$y\in G \uparrow$$, then $$xyx=y^{-1}$$; (iii) if $$G \downarrow\neq\emptyset$$, then $$G \uparrow$$ is Abelian. The most important result of the paper is the following theorem: Let $$G$$ be a half lattice-ordered group; then there exists a chain $$S$$ such that $$G$$ is isomorphic to a sub-structure of $$M(S)$$. This generalizes the well-known Holland’s theorem concerning the representation of a lattice-ordered group as group of automorphisms of some chain. Further the authors deal with subgroups of a half lattice- ordered group and they establish a characterization of all half-ordered groups $$G$$ such that $$G \uparrow$$ is isomorphic to a given lattice- ordered group $$H$$.

##### MSC:
 06F15 Ordered groups
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