We consider linear isotopes of commutative groups, i.e., central quasigroups and study the invertibility and orthogonality conditions. It turns out that it is sufficient to study these conditions solely for the unitary isotopes, i.e., for isotopes, which have an idempotent. We establish criteria for the possession of each invertibility property (inverse property, crossed inverse property, and mirroring) for unitary central and matrix quasigroups. In particular, for matrices of the second order, we describe the corresponding matrix quasigroups over the fields of characteristics 2 and 3. The orthogonality criteria are obtained for matrix quasigroups with the indicated invertibility properties.
Similar content being viewed by others
References
V. D. Belousov, Fundamentals of the Theory of Quasigroups and Loops [in Russian], Nauka, Moscow (1967).
V. D. Belousov and V. D. Tsurkan, “Crossed-inverse quasigroups (CI-quasigroups),” Izv. Vyssh. Uchebn. Zaved., Mat., No. 3(82), 21–27 (1969).
F. N. Sokhatskii, “On isotopes of groups. II,” Ukr. Mat. Zh., 47, No. 12, 1692–1703 (1995); English translation: Ukr. Math. J., 47, No. 12, 1935–1948 (1995); https://doi.org/10.1007/BF01060967.
A. V. Lutsenko, “Classification of group isotopes according to their inverse properties,” Prykl. Probl. Mekh. Mat., Issue 18, 48–61 (2020); https://doi.org/10.15407/apmm2020.18.48-61.
H. O. Pflugfelder, “Quasigroups and loops: introduction,” in: Sigma Series in Pure Mathematics, Vol. 7, Heldermann Verlag, Berlin (1990).
F. M. Sokhatsky, “Factorization of operations of medial and Abelian algebras,” Visn. Donets’k. Nats. Univ. Ser. A: Pryrodn. Nauky, No. 1-2, 84–96 (2017); https://doi.org/10.31558/1817-2237.2017.1-2.7.
F. M. Sokhatsky and I. V. Fryz, “Invertibility criterion of composition of two multiary quasigroups,” Comment. Math. Univ. Carolin., 53, No. 3, 429–445 (2012).
F. M. Sokhatsky, H. V. Krainichuk, and V. A. Sydoruk, “Semi-lattice of varieties of quasigroups with linearity,” Algebra Discrete Math., 31, No. 2, 261–285 (2021); https://doi.org/10.12958/adm1748.
F. M. Sokhatsky and A. V. Lutsenko, “The bunch of varieties of inverse property quasigroups,” Visn. Donets’k. Nats. Univ. Ser. A: Pryrodn. Nauky, No. 1-2, 56–69 (2018); https://doi.org/10.31558/1817-2237.2018.1-2.4.
F. Sokhatsky and A. Lutsenko, “Classification of quasigroups according to directions of translations. I,” Comment. Math. Univ. Carolin., 61, No. 4, 567–579 (2020).
F. Sokhatsky and A. Lutsenko, “Classification of quasigroups according to directions of translations. II,” Comment. Math. Univ. Carolin., 62, No. 3, 309–323 (2021).
F. Sokhatskyj and P. Syvakivskyj, “On linear isotopes of cyclic groups,” Quasigroups Relat. Syst., 1, No. 1(1), 66–76 (1994).
B. V. Zabavsky, O. V. Domsha, and O. M. Romaniv, “Clear rings and clear elements,” Mat. Stud., 55, No. 1, 3–9 (2021); https://doi.org/10.30970/ms.55.1.3-9.
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 64, No. 4, pp. 5–17, October–December, 2021.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Sokhatsky, F.M., Lutsenko, A.V. & Fryz, I.V. Construction of Quasigroups with Invertibility Properties. J Math Sci 279, 115–132 (2024). https://doi.org/10.1007/s10958-024-06999-0
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-024-06999-0