Parastrophic-orthogonal ternary medial quasigroups with 3 and 4 distinct parastrophes
DOI:
https://doi.org/10.31652/3041-1955-2026-03-01-07Keywords:
ternary quasigroup, group isotope, medial quasigroup, parastrophe, (strongly) orthogonal quasigroups, totally parastrophic-orthogonal (top) quasigroupAbstract
In the article, we study parastrophic-orthogonal ternary quasigroups: namely, group isotopes which have 3 and 4 distinct parastrophes. The necessary and sufficient conditions for ternary medial quasigroups with 3 and 4 distinct parastrophes to be totally parastrophic-orthogonal are proved. The conditions under which these quasigroups are strongly parastrophic-orthogonal are described. Thus, some methods of constructing orthogonal and strongly orthogonal ternary quasigroups are obtained.
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McLeish, M. (1979) . On the number of conjugates of n-ary quasigroups, Can. J. Math. XXXI (3), 637-654. https://doi.org/10.4153/CJM-1979-064-6
McLeish M. (1980) . On the Existence of Ternary Quasi-Groups with Two or Eight Conjugacy Classes, Journal of Combinatorial Theory, Series A, 29, 199-211. https://doi.org/10.1016/0097-3165(80)90009-6
Sokhatsky, F., Pirus, Ye. (2018). Classification of ternary quasigroups according to their parastrophic symmetry groups, I, Visnyk DonNu. Series A: Natural Sciences, 1-2, 70-82. https://doi.org/10.31558/1817-2237.2018.1-2.5
Pirus, Ye. (2019) . Classification of ternary quasigroups according to their parastrophic symmetry groups, II, Visnyk DonNu. Series A: Natural Sciences, 1-2, 66-75. https://doi.org/10.31558/1817-2237.2019.1-2.9
Fryz I., Sokhatsky, F. (2022). Construction of medial ternary self-orthogonal quasigroups, Bul. Acad. Stiinte Repub. Mold. Mat., 3 (100), 41-55. https://doi.org/10.56415/basm.y2022.i3.p41
Belyavskaya, G.B., Popovich, T.V. (2010) . Totally conjugate orthogonal quasigroups and complete graphs, J. Math. Sci., 185 (2), 184-191. https://doi.org/10.1007/s10958-012-0907-z
Sokhatsky, F. (2016). Parastrophic symmetry in quasigroup theory, Visnyk DonNu. Series A: Natural Sciences, 1-2, 70-83.
Sokhatsky, F. (2017). Factorization of operations of medial and abelian algebras, Visnyk DonNY. Series A: Natural Sciences, 1-2, 84-96. https://doi.org/10.31558/1817-2237.2017.1-2.7
Belyavskaya, G., Mullen, G.L. (2006). Strongly orthogonal and uniformly orthogonal many-placed operations, Algebra Discrete Math., 5, 1, 1-17.
Pirus, Ye. (2020) On ternary top-quasigroups whose group of parastrophic symmetry is D_8. Book of Abstracts of International mathematical conference dedicated to the 60th anniversary of the department of algebra and mathematical logic of Taras Shevchenko National University of Kyiv (Kyiv, Ukraine, July 14-17, 2020), 64.
Rotari, T. (2024). On ternary quasigoups with exactly three distinct and orthogonal parastrophes. e-Book of Abstracts of the 31nd International Conference on Applied and Industrial Mathematics (Oradea, Romania, September 19-22, 2024), 53-54.
Pirus, Ie. (2015). About parastrophic orthogonality of medial ternary quasigroups. Abstracts of X International Algebraic Conference in Ukraine dedicated to the 70th anniversary of Yu.A. Drozd (Odesa, Ukraine, August 20-27, 2015), 87.
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Copyright (c) 2026 Ірина Фриз, Євген Пірус
This work is licensed under a Creative Commons Attribution 4.0 International License.
