ArticleOriginal scientific text

Title

On the increasing solutions of the translation equation

Authors 1

Affiliations

  1. Institute of Mathematics, Pedagogical University of Rzeszów, Rejtana 16a, 35-310 Rzeszów, Poland

Abstract

Let M be a non-empty set endowed with a dense linear order without the smallest and greatest elements. Let (G,+) be a group which has a non-trivial uniquely divisible subgroup. There are given conditions under which every solution F: M×G → M of the translation equation is of the form F(a,x)=f-1(f(a)+c(x)) for a ∈ M, x ∈ G with some non-trivial additive function c: G → ℝ and a strictly increasing function f: M → ℝ such that f(M) + c(G) ⊂ f(M). In particular, a problem of J. Tabor is solved.

Keywords

translation equation, linear order, increasing function, additive function

Bibliography

  1. U. Abel, Sur les groupes d'itération monotones, Publ. Math. Debrecen 29 (1982), 65-71.
  2. J. Aczél, L. Kalmár et J. G. Mikusiński, Sur l'équation de translation, Studia Math. 12 (1951), 112-116.
  3. A. Grzegorczyk and J. Tabor, Monotonic solutions of the translation equation, Ann. Polon. Math. 43 (1983), 253-260.
  4. J. Tabor, Characterization of mixed iteration groups, to appear.
Pages:
207-214
Main language of publication
English
Received
1991-11-16
Accepted
1995-11-08
Published
1996
Exact and natural sciences