On the increasing solutions of the translation equation

Volume 64 / 1996

Janusz Brzdęk Annales Polonici Mathematici 64 (1996), 207-214 DOI: 10.4064/ap-64-3-207-214


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.


  • Janusz Brzdęk

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