| Title:
|
Solution of distributive-like quasigroup functional equations (English) |
| Author:
|
Sokhatsky, Fedir M. |
| Author:
|
Krainichuk, Halyna V. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
53 |
| Issue:
|
3 |
| Year:
|
2012 |
| Pages:
|
447-459 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
quasigroup |
| Keyword:
|
functional equation |
| Keyword:
|
distributive quasigroup |
| Keyword:
|
distributive-like functional equation |
| Keyword:
|
quasigroup solution |
| Keyword:
|
solution set |
| Keyword:
|
quasigroup identity |
| Keyword:
|
parastrophic equivalence |
| MSC:
|
05B15 |
| MSC:
|
20N05 |
| idZBL:
|
Zbl 1265.39041 |
| idMR:
|
MR3017842 |
| . |
| Date available:
|
2012-08-31T11:42:34Z |
| Last updated:
|
2014-10-06 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/142936 |
| . |
| Reference:
|
[1] Aczél J.: Lectures on Functional Equations and Their Applications.Academic Press, New York, London, 1966. MR 0208210 |
| Reference:
|
[2] Aczél J., Belousov V.D., Hosszú M.: Generalized associativity and bisymmetry on quasigroups.Acta. Math. Acad. Sci. Hungar. 11 (1960), no. 12-2, 127-136. MR 0140600, 10.1007/BF02020630 |
| Reference:
|
[3] Belousov V.D.: Associative system of quasigroups.Uspekhi Mat. Nauk, (1958), 13, 3(81), 243 (in Russian). |
| Reference:
|
[4] Belousov V.D.: Cross isotopy of quasigroup.Quasigroups and Their Systems, Shtiintsa, Kishinev (1990), 14–20 (in Russian). |
| Reference:
|
[5] Belousov V.D.: Some remarks on the functional equation of generalized distributivity.Aequationes Math. 1 (1968), no. 1–2, 54–65. Zbl 0157.46402, MR 0228610, 10.1007/BF01817557 |
| Reference:
|
[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). MR 0256328 |
| Reference:
|
[7] Koval' R.F.: On a functional equation with a group isotopy property.Bul. Acad. Stiinte Repub. Mold. Mat. 2005, no. 2, 65–71. MR 2190739 |
| Reference:
|
[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). |
| Reference:
|
[9] Krapež A., Živković D.: Parastrophically equivalent quasigroup equations.Publ. Inst. Math. (Beograd) (N.S.), 87(101), (2010), 39–58. MR 2642000, 10.2298/PIM1001039K |
| Reference:
|
[10] Krapež A., Simić S.K., Tošić D.V.: Parastrophically uncancellable quasigroup equations.Aequationes Math. 79 (2010), 261-280. Zbl 1217.39032, MR 2665535, 10.1007/s00010-010-0016-3 |
| Reference:
|
[11] Sokhatsky F.M.: On classification of functional equations on quasigroups.Ukrainian Math. J. 56 (2004), no. 4, 1259–1266 (in Ukrainian). MR 2133028 |
| Reference:
|
[12] 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. |
| . |
