On dilations and contractions in Riesz groups. (English) Zbl 0704.06009

Jasem, Milan

Čas. Pěstování Mat. 115, No. 2, 134-141 (1990).

For n,m\(\in N\), a bijection f: \(G\to G\) is called an (m,n)-transposition in a partially ordered group G if \(m| f(x)-f(y)| =n| x- y|\) for each x,y\(\in G\). If \(m<n\) \((m>n)\), then an (m,n)- transposition in an isolated partially ordered group is called a dilation (contraction). The main result establishes the relations between the (m,n)-transpositions in an isolated abelian Riesz group G and the direct decompositions of G. Further, it is shown that (m,n)-transpositions in G preserve certain convex subsets of G.

Reviewer: S.A.Gurchenkov

MSC:

06F15	Ordered groups
06F20	Ordered abelian groups, Riesz groups, ordered linear spaces

Keywords:

isometry; dilation; contraction; Riesz group; direct decompositions; convex subsets