Geodesic
On Bruck-Belousov Problem
Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 3 (2005), pp. 123-140.

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In this paper on the language of subgroups of the multiplication group of a quasigroup (of the associated group of a quasigroup) necessary and sufficient conditions of normality of congruences of a left (right) loop are given. These conditions can be considered as a partial answer to the problem posed in books of R. H. Bruck and V. D. Belousov about conditions of normality of all congruences of quasigroups. Results on the regularity of congruences of quasigroups and the behavior of quasigroup congruences by isotopy are given. 
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Victor Shcherbacov. On Bruck-Belousov Problem. Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 3 (2005), pp. 123-140. https://geodesic-test.mathdoc.fr/item/BASM_2005_3_a10/

[1] Albert A. A., “Quasigroups, I”, Trans. Amer. Math. Soc., 54 (1943), 507–519 | DOI | MR | Zbl

[2] Albert A. A., “Quasigroups, II”, Trans. Amer. Math. Soc., 55 (1944), 401–419 | DOI | MR | Zbl

[3] Bates G. E., Kiokemeister F., “A note on homomorphic mappings of quasigroups into multiplicative systems”, Bull. Amer. Math. Soc., 54 (1948), 1180–1185 | DOI | MR | Zbl

[4] Beglaryan V. A., To theory of homomorphisms in quasigroups, Thesis of Ph. Degree, Kishinev, 1981 (in Russian)

[5] Belousov V. D., Foundations of the theory of quasigroups and loops, Nauka, Moscow, 1967 | MR

[6] Belousov V. D., Elements of the Quasigroup Theory: A special course, Kishinev State University Press, Kishinev, 1981 (in Russian) | MR | Zbl

[7] Belousov V. D., “Inverse loops”, Mat. Issled., 95, Shtiintsa, Kishinev, 1987, 3–22 (in Russian) | MR

[8] Belousov V. D., “Inverse loops”, Mat. Issled., 95, Shtiintsa, Kishinev, 1987, 3–22 (in Russian) | MR

[9] Belyavskaya G. B., “Direct decompositions of quasigroups”, Mat. Issled., 95, Shtiintsa, Kishinev, 1987, 23–38 (in Russian) | MR

[10] Birkhoff G., Lattice theory, Nauka, Moscow, 1984 (in Russian) | MR

[11] Bruck R. H., A Survey of Binary Systems, Springer Verlag, Berlin, 1958 | MR | Zbl

[12] Clifford A. H., Preston G. B., The algebraic thery of semigroups, V. 1, Mir, Moscow, 1972 (in Russian) | MR

[13] Cohn P. M., Universal Algebra, Harper Row, New York, 1965 | MR | Zbl

[14] Kargapolov M. I., Merzlyakov Yu. I., Foundations of the Group Theory, Nauka, Moskow, 1977 (in Russian) | MR | Zbl

[15] Karklin'š B. B., Karklin' V. B., “Inverse loops”, Nets and groups, Mat. Issled., 39, Shtiintsa, Kishinev, 1976, 87–101 (in Russian) | MR

[16] Keedwell A. D., Shcherbacov V. A., “Construction and properties of $(r,s,t)$-inverse quasigroups, I”, Discrete Math., 266:1–3 (2003), 275–291 | DOI | MR | Zbl

[17] Keedwell A. D., Shcherbacov V. A., “Quasigroups with an inverse property and generalized parastrophic identities”, Quasigroups and related systems, 13 (2005), 109–124 | MR | Zbl

[18] Kepka T., Němec P., “$T$-quasigroups, II”, Acta Universitatis, Carolinae Math. et Physica, 12:2 (1971), 31–49

[19] Kurosh A. G., The Group Theory, Nauka, Moscow, 1967 (in Russian) | Zbl

[20] Kurosh A. G., Lectures on General Algebra, Gos. Izd. Fiz.-mat. Lit., Moscow, 1962 (in Russian) | MR

[21] Mal'tsev A. I., “To general theory of algebraic systems”, Mat. sbornik, 35 (1954), 3–20 (in Russian)

[22] Parovichenko I. I., The Theory of Operations over Sets, Shtiintsa, Kishinev, 1981 | MR

[23] Pflugfelder H. O., Quasigroups and Loops: Introduction, Heldermann Verlag, Berlin, 1990 | MR

[24] Riguet J., “Relations binares, fermetures, correspondances de Galois”, Bull. Soc. Math. France, 76 (1948), 114–155 | MR

[25] Thurston H. A., “Equivivalences and mappings”, Proc. London Math. Soc., 3:2 (1952), 175–182 | DOI | MR | Zbl

[26] Thurston H. A., “Certain congruences on quasigroups”, Proc. Amer. Math. Soc., 3 (1952), 10–12 | DOI | MR | Zbl

[27] Shcherbacov V. A., On congruences of quasigroups, Reg. in VINITI 01.08.90, No 4413-B90, 1990, Moscow, in Russian

[28] Shcherbacov V. A., On automorphism groups and congruences of quasigroups, Thesis of Ph. Degree, Kishinev, 1991 (in Russian)

[29] Shchukin K. K., Action of a group on a quasigroup, Kishinev State University Press, Kishinev, 1985 (in Russian)

[30] Smith J. D. H., Mal'cev Varieties, Lecture Notes in Mathematics, 554, 1976 | MR | Zbl