Abstract
AbstractIn the paper weak isometries in directed groups are investigated. It is proved that for every weak isometryfin a directed groupGthe relationf(UL(x,y) ∩LU(x,y)) =UL(f(x),f(y)) ∩LU(f(x),f(y)) is valid for eachx,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 groupGare directly related to subdirect decompositions of the subgroupG2= {2x;x∈G} ofG.