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Isometries in Riesz groups. (English) Zbl 0603.06007
Let G be a po-group and f a bijection $$G\to G$$. f is called an isometry in G if $$| a-b| =| f(a)-f(b)|$$ for each a,b$$\in G$$. An isometry f in G is said to be a 0-isometry if $$f(0)=0$$. In the paper isometries in abelian Riesz groups are studied. Main results are obtained. 1. For each 0-isometry f in G there exists a direct decomposition $$G=A\times B$$ such that $$f(x)=x_ A-x_ B$$ for each $$x\in G$$. $$(x_ A$$ and $$x_ B$$ are respectively the projections of x into A and B.) Conversely, if $$G=P\times Q$$ is a direct decomposition of G and if we put $$g(x)=x_ P-x_ Q$$ for each $$x\in G$$, then g is a 0-isometry. 2. If g is an isometry in G and x,y$$\in G$$, then $g[U(L(x,y))\cap L(U(x,y))]=U[L(g(x),g(y))\cap L[U(g(x),g(y))].$ 3. $$\emptyset \neq H$$ ($$\subseteq G)$$ is a directed convex subgroup of G iff f(H) is a directed convex subgroup of G. Examples are given which prove that the property to be directed or convex cannot be dropped in 3.
Reviewer: F.Šik

##### MSC:
 06F20 Ordered abelian groups, Riesz groups, ordered linear spaces 06F15 Ordered groups
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##### References:
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