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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

06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
06F15 Ordered groups
Full Text: EuDML
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