# On the increasing solutions of the translation equation

• Volume: 64, Issue: 3, page 207-214
• ISSN: 0066-2216

top

## Abstract

top
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 $F\left(a,x\right)={f}^{-1}\left(f\left(a\right)+c\left(x\right)\right)$ 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.

## How to cite

top

Janusz Brzdęk. "On the increasing solutions of the translation equation." Annales Polonici Mathematici 64.3 (1996): 207-214. <http://eudml.org/doc/270001>.

@article{JanuszBrzdęk1996,
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.},
author = {Janusz Brzdęk},
journal = {Annales Polonici Mathematici},
keywords = {translation equation; linear order; increasing function; additive function; dense linear order; group; uniquely divisible subgroup},
language = {eng},
number = {3},
pages = {207-214},
title = {On the increasing solutions of the translation equation},
url = {http://eudml.org/doc/270001},
volume = {64},
year = {1996},
}

TY - JOUR
AU - Janusz Brzdęk
TI - On the increasing solutions of the translation equation
JO - Annales Polonici Mathematici
PY - 1996
VL - 64
IS - 3
SP - 207
EP - 214
AB - 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.
LA - eng
KW - translation equation; linear order; increasing function; additive function; dense linear order; group; uniquely divisible subgroup
UR - http://eudml.org/doc/270001
ER -

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.