# On the increasing solutions of the translation equation

Annales Polonici Mathematici (1996)

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

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topJanusz 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 -

## References

top- [1] U. Abel, Sur les groupes d'itération monotones, Publ. Math. Debrecen 29 (1982), 65-71. Zbl0505.39005
- [2] J. Aczél, L. Kalmár et J. G. Mikusiński, Sur l'équation de translation, Studia Math. 12 (1951), 112-116. Zbl0042.34401
- [3] A. Grzegorczyk and J. Tabor, Monotonic solutions of the translation equation, Ann. Polon. Math. 43 (1983), 253-260. Zbl0537.39008
- [4] J. Tabor, Characterization of mixed iteration groups, to appear. Zbl1167.26003

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