caa a22 4500 465824951 CHVBK 20180323112146.0 cr unu---uuuuu 170327e19900201xx s 000 0 eng 10.1007/BF01833938 doi (NATIONALLICENCE)springer-10.1007/BF01833938 On Tabor's problem concerning a certain quasi-ordering of iterative roots of functions [Elektronische Daten] [Jaroslav Beránek, Jan Chvalina] Summary: Using the Isaacs-Zimmermann's theory of iterative roots of functions, we prove a theorem concerning the problemP 250 posed by J. Tabor: "Letf: E → E be a given mapping. Denote byF the set of all iterative roots off. InF we define the following relation:ϕ ≦ ψ if and only ifϕ is an iterative root ofψ. The relation is obviously reflexive and transitive. The question is: Is it also antisymmetric? If we consider iterative roots of a monotonic function the answer is ‘yes'. But in general the question is open.” Here we prove that there exists a three-element decomposition {Φ i ;i = 1, 2, 3} of the setE E with blocks Φi of the same cardinality 2cardE such that the functions from Ф1 do not possess any proper iterative root, the quasi-ordering ≦ is not antisymmetric onF(f) for anyf ∈ Φ2, and ≦ is an ordering onF(f) for anyf ∈ Ф3. Iff is a strictly increasing continuous self-bijection ofE, then the relation ≦ is an ordering onF(f) ifff is different from the identity mapping of the setE. Birkhäuser Verlag, 1990 Primary 26A18 nationallicence Secondary 39B05, 06A10 nationallicence Beránek Jaroslav Department of Mathematics, Pedagogical Faculty of J. E. Purkyně University, Poříčí 31, ČSR-603 00, Brno, Czechoslovakia aut Chvalina Jan Department of Mathematics, Pedagogical Faculty of J. E. Purkyně University, Poříčí 31, ČSR-603 00, Brno, Czechoslovakia aut aequationes mathematicae Birkhäuser-Verlag 39/1(1990-02-01), 1-5 0001-9054 39:1<1 1990 39 10 https://doi.org/10.1007/BF01833938 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/BF01833938 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Beránek Jaroslav Department of Mathematics, Pedagogical Faculty of J. E. Purkyně University, Poříčí 31, ČSR-603 00, Brno, Czechoslovakia aut NATIONALLICENCE 700 1- Chvalina Jan Department of Mathematics, Pedagogical Faculty of J. E. Purkyně University, Poříčí 31, ČSR-603 00, Brno, Czechoslovakia aut NATIONALLICENCE 773 0- aequationes mathematicae Birkhäuser-Verlag 39/1(1990-02-01), 1-5 0001-9054 39:1<1 1990 39 10 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer