Solution of distributive-like quasigroup functional equations

Fedir M. Sokhatsky; Halyna V. Krainichuk

Commentationes Mathematicae Universitatis Carolinae (2012)

  • Volume: 53, Issue: 3, page 447-459
  • ISSN: 0010-2628

Abstract

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We are investigating quasigroup functional equation classification up to parastrophic equivalence [Sokhatsky F.M.: On classification of functional equations on quasigroups, Ukrainian Math. J. 56 (2004), no. 4, 1259–1266 (in Ukrainian)]. If functional equations are parastrophically equivalent, then their functional variables can be renamed in such a way that the obtained equations are equivalent, i.e., their solution sets are equal. There exist five classes of generalized distributive-like quasigroup functional equations up to parastrophic equivalence [Sokhatsky F.M.: On classification of distributive-like functional equations, Book of Abstracts of the 8 t h International Algebraic Conference in Ukraine, July 5–12 (2011), Lugansk, Ukraine, p. 79]. In the article, we find the solution sets of four generalized distributive-like quasigroup functional equations of different classes. In consequence, we solve one of the equations on topological quasigroup operations, defined on arbitrary topological space as well as on the space of real numbers with the natural topology. The fifth class contains the generalized left distributivity functional equation. V.D. Belousov [Some remarks on the functional equation of generalized distributivity, Aequationes Math. 1 (1968), no. 1–2, 54–65] described only a subset of its solution set. The set of all solutions still remains an open problem in the quasigroup theory and in the functional equation theory.

How to cite

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Sokhatsky, Fedir M., and Krainichuk, Halyna V.. "Solution of distributive-like quasigroup functional equations." Commentationes Mathematicae Universitatis Carolinae 53.3 (2012): 447-459. <http://eudml.org/doc/246658>.

@article{Sokhatsky2012,
abstract = {We are investigating quasigroup functional equation classification up to parastrophic equivalence [Sokhatsky F.M.: On classification of functional equations on quasigroups, Ukrainian Math. J. 56 (2004), no. 4, 1259–1266 (in Ukrainian)]. If functional equations are parastrophically equivalent, then their functional variables can be renamed in such a way that the obtained equations are equivalent, i.e., their solution sets are equal. There exist five classes of generalized distributive-like quasigroup functional equations up to parastrophic equivalence [Sokhatsky F.M.: On classification of distributive-like functional equations, Book of Abstracts of the $8^\{th\}$ International Algebraic Conference in Ukraine, July 5–12 (2011), Lugansk, Ukraine, p. 79]. In the article, we find the solution sets of four generalized distributive-like quasigroup functional equations of different classes. In consequence, we solve one of the equations on topological quasigroup operations, defined on arbitrary topological space as well as on the space of real numbers with the natural topology. The fifth class contains the generalized left distributivity functional equation. V.D. Belousov [Some remarks on the functional equation of generalized distributivity, Aequationes Math. 1 (1968), no. 1–2, 54–65] described only a subset of its solution set. The set of all solutions still remains an open problem in the quasigroup theory and in the functional equation theory.},
author = {Sokhatsky, Fedir M., Krainichuk, Halyna V.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {quasigroup; functional equation; distributive quasigroup; distributive-like functional equation; quasigroup solution; solution set; quasigroup identity; parastrophic equivalence; distributive quasigroup; distributive-like functional equation; quasigroup solution; parastrophic equivalence; topological quasigroup},
language = {eng},
number = {3},
pages = {447-459},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Solution of distributive-like quasigroup functional equations},
url = {http://eudml.org/doc/246658},
volume = {53},
year = {2012},
}

TY - JOUR
AU - Sokhatsky, Fedir M.
AU - Krainichuk, Halyna V.
TI - Solution of distributive-like quasigroup functional equations
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2012
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 53
IS - 3
SP - 447
EP - 459
AB - We are investigating quasigroup functional equation classification up to parastrophic equivalence [Sokhatsky F.M.: On classification of functional equations on quasigroups, Ukrainian Math. J. 56 (2004), no. 4, 1259–1266 (in Ukrainian)]. If functional equations are parastrophically equivalent, then their functional variables can be renamed in such a way that the obtained equations are equivalent, i.e., their solution sets are equal. There exist five classes of generalized distributive-like quasigroup functional equations up to parastrophic equivalence [Sokhatsky F.M.: On classification of distributive-like functional equations, Book of Abstracts of the $8^{th}$ International Algebraic Conference in Ukraine, July 5–12 (2011), Lugansk, Ukraine, p. 79]. In the article, we find the solution sets of four generalized distributive-like quasigroup functional equations of different classes. In consequence, we solve one of the equations on topological quasigroup operations, defined on arbitrary topological space as well as on the space of real numbers with the natural topology. The fifth class contains the generalized left distributivity functional equation. V.D. Belousov [Some remarks on the functional equation of generalized distributivity, Aequationes Math. 1 (1968), no. 1–2, 54–65] described only a subset of its solution set. The set of all solutions still remains an open problem in the quasigroup theory and in the functional equation theory.
LA - eng
KW - quasigroup; functional equation; distributive quasigroup; distributive-like functional equation; quasigroup solution; solution set; quasigroup identity; parastrophic equivalence; distributive quasigroup; distributive-like functional equation; quasigroup solution; parastrophic equivalence; topological quasigroup
UR - http://eudml.org/doc/246658
ER -

References

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  1. Aczél J., Lectures on Functional Equations and Their Applications, Academic Press, New York, London, 1966. MR0208210
  2. Aczél J., Belousov V.D., Hosszú M., 10.1007/BF02020630, Acta. Math. Acad. Sci. Hungar. 11 (1960), no. 12-2, 127-136. MR0140600DOI10.1007/BF02020630
  3. Belousov V.D., Associative system of quasigroups, Uspekhi Mat. Nauk, (1958), 13, 3(81), 243 (in Russian). 
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  5. Belousov V.D., 10.1007/BF01817557, Aequationes Math. 1 (1968), no. 1–2, 54–65. Zbl0157.46402MR0228610DOI10.1007/BF01817557
  6. Bourbaki N., General Topology. Topological Groups. Numbers and Related to them Groups and Spaces, Nauka, Moscow, 1969, 392 pp. (Russian, translated from the French). MR0256328
  7. Koval' R.F., On a functional equation with a group isotopy property, Bul. Acad. Stiinte Repub. Mold. Mat. 2005, no. 2, 65–71. MR2190739
  8. Krainichuk H.V., Sokhatsky F.M., Solving of some functional equations having invertible binary functions, Academ. Ya.S. Pidstryhach Conf. of Young Scientists “Modern problems of Math. and Mech”, Lviv Ivan Franko National University, Lviv, 2009, pp. 158–159 (in Ukrainian). 
  9. Krapež A., Živković D., 10.2298/PIM1001039K, Publ. Inst. Math. (Beograd) (N.S.), 87(101), (2010), 39–58. MR2642000DOI10.2298/PIM1001039K
  10. Krapež A., Simić S.K., Tošić D.V., 10.1007/s00010-010-0016-3, Aequationes Math. 79 (2010), 261-280. Zbl1217.39032MR2665535DOI10.1007/s00010-010-0016-3
  11. Sokhatsky F.M., On classification of functional equations on quasigroups, Ukrainian Math. J. 56 (2004), no. 4, 1259–1266 (in Ukrainian). MR2133028
  12. Sokhatsky F.M., On classification of distributive-like functional equations, Book of Abstracts of the International Algebraic Conference in Ukraine, July 5–12 (2011), Lugansk, Ukraine, p. 79. 

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