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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:
 [1] Л. Фу\kappa с: Частично упроядоченные алгебраические системы. Москва 1965. · Zbl 1099.01519 [2] Я. Якубик: Прямые разложения частично упроядоченных групп. Чехосл. матем. ж. 10 (1960), 231-243; 11 (1961), 490-515. · Zbl 1004.90500 [3] J. Jakubík: Isometries of lattice ordered groups. Czech. Math. J. 30 (105) (1980), 142-152. · Zbl 0436.06013 [4] J. Jakubík: On isometries of non-abelian lattice ordered groups. Math. Slovaca 31 (1981), 171-175. [5] J. Jakubík M. Kolibiar: Isometries of multilattice groups. Czech. Math. J. 33 (1983), 602-612. · Zbl 0538.06018 [6] W. B. Powell: On isometries in abelian lattice ordered groups. Preprint, Oklahoma State University. · Zbl 0614.06012 [7] J. Rachůnek: Isometries in ordered groups. Czech. Math. J. 34 (109) (1984), 334-341. · Zbl 0558.06020 [8] K. L. Swamy: Isometries in autometrized lattice ordered groups. Algebra Univ. 8 (1977), 58-64. · Zbl 0457.06015 [9] K. L. Swamy: Isometries in autometrized lattice ordered groups, II. Math. Seminar Notes Kobe Univ. 5 (1977), 211-214. · Zbl 0457.06015
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