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Isometries in ordered groups. (English) Zbl 0558.06020

For abelian lattice-ordered groups G, the absolute value \(d(a,b)=| a- b| =(a-b)\vee (b-a)\) can be considered as a kind of metric on G with values in G, and one may consider isometries of G with respect to this metric. These investigations have been begun by K. L. N. Swamy [Math. Ann. 154, 406-412 (1964; Zbl 0128.254)]. J. Jakubík has extended these notions to the non-abelian case [Math. Slovaca 31, 171-175 (1981; Zbl 0457.06014)]. In this paper the author goes away from lattice-ordered groups. In any ordered group he defines \(| a|\) to be the set of all common upper bounds of a and -a, and a generalized ”metric” is defined by \(d(a,b)=| a-b|\). One cannot expect, that this ”metric” has reasonable properties in general. In the paper under review the author exhibits a fairly large class of abelian Riesz groups for which the generalized ”metric” behaves very well and he is able to deal with ”isometries” in a very reasonable way.
Reviewer: K.Keimel

MSC:

06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
54E35 Metric spaces, metrizability
06F30 Ordered topological structures
54E40 Special maps on metric spaces
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References:

[1] A. Bigard K. Keimel S. Wolfenstein: Groupes et Anneaux Réticulés. Berlin-Heidelberg- New York, 1977.
[2] L. Fuchs: Riesz groups. Ann. Sc. Norm. Super. Pisa, Ser. III, Vol. XIX, Fase. I (1965), 1-34. · Zbl 0125.28703
[3] L. Fuchs: Partially ordered algebraic systems. (Russian), Moscow, 1965. · Zbl 0137.02001
[4] J. Jakubík: Isometries of lattice ordered groups. Czech. Math. J. 30 (105) (1980), 142-152. · Zbl 0436.06013
[5] J. Jakubík: On isometries of non-abelian lattice ordered groups. Math. Slovaca 31 (1981), 171-175.
[6] K. L. N. Swamy: Autometrized lattice ordered groups I. Math. Ann. 154 (1964), 406-412. · Zbl 0128.25403 · doi:10.1007/BF01375523
[7] K. L. N. Swamy: Dually residuated lattice ordered semigroups II. Math. Ann. 160 (1965), 64-71. · Zbl 0138.02104 · doi:10.1007/BF01364335
[8] K. L. N. Swamy: Isometries in autometrized lattice ordered groups. Alg. Univ. 8 (1978), 59-64. · Zbl 0409.06007 · doi:10.1007/BF02485370
[9] K. L. N. Swamy: Isometries in autometrized lattice ordered groups II. Math. Sem. Notes Kobe Univ. 5 (1977), 211-214. · Zbl 0457.06015
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