# 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

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topSokhatsky, 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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- Belousov V.D., 10.1007/BF01817557, Aequationes Math. 1 (1968), no. 1–2, 54–65. Zbl0157.46402MR0228610DOI10.1007/BF01817557
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- Krapež A., Živković D., 10.2298/PIM1001039K, Publ. Inst. Math. (Beograd) (N.S.), 87(101), (2010), 39–58. MR2642000DOI10.2298/PIM1001039K
- 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
- Sokhatsky F.M., On classification of functional equations on quasigroups, Ukrainian Math. J. 56 (2004), no. 4, 1259–1266 (in Ukrainian). MR2133028
- 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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