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On weak isometries in directed groups

Authors
  • Jasem, Milan
Type
Published Article
Journal
Mathematica Slovaca
Publisher
De Gruyter
Publication Date
Oct 05, 2019
Volume
69
Issue
5
Pages
989–998
Identifiers
DOI: 10.1515/ms-2017-0283
Source
De Gruyter
Keywords
License
Yellow

Abstract

In the paper weak isometries in directed groups are investigated. It is proved that for every weak isometry f in a directed group G the relation f(UL(x, y) ∩ LU(x, y)) = UL(f(x), f(y)) ∩ LU(f(x), f(y)) is valid for each x, y ∈ G. The notions of an orthogonality of two elements and of a subgroup symmetry in directed groups are introduced and it is shown that each weak isometry in a 2-isolated directed group or in an abelian directed group is a composition of a subgroup symmetry and a right translation. It is also proved that stable weak isometries in a 2-isolated abelian directed group G are directly related to subdirect decompositions of the subgroup G2 = {2x; x ∈ G} of G.

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