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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 \(H\).

06F15 Ordered groups
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