On orthogonality of binary operations and squares

Gary L. Mullen ; Victor A. Shcherbacov

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 2 (2005), pp. 3-42.

Résumé

Orthogonality of a pair of binary groupoids, left quasigroups and quasigroups from some points of view is studied. Necessary and sufficient conditions of orthogonality of a finite quasigroup and any its parastrophe (conjugate quasigroup in other terminology), including ones in language of quasi-identities, are given. New concept of gisotopy, which generalizes the concept of isotopy, is defined. There is information on quasigroups with self-orthogonal conjugates.

Bibliographie

[1] Barlotti A., Strambach K., “The geometry of binary systems”, Adv. in Math., 49 (1983), 1–105 | DOI | MR | Zbl

[2] Bektenov A. S., Yakubov T., “Systems of orthogonal $n$-ary operations”, Izvestiya AN MSSR, Ser. fiz.-tekh. i mat. nauk, 1974, no. 3, 7–14 (in Russian) | MR | Zbl

[3] Belousov V. D., “On properties of binary operations”, Uchenye zapiski Beltskogo ped. instituta, 1960, no. 5, 9–28 (in Russian)

[4] Belousov V. D., Foundations of the Theory of Quasigroups and Loops, Nauka, Moscow, 1967 (in Russian) | MR

[5] Belousov V. D., “Systems of orthogonal operations”, Mat. sbornik, 77(119):1 (1968), 38–58 (in Russian) | MR | Zbl

[6] Belousov V. D., “On group associated with a quasigroup”, Mat. issled., 4, no. 3, RIO AN MSSR, Kishinev, 1969, 21–39 (in Russian) | MR

[7] Belousov V. D., Algebraic nets and quasigroups, Shtiintsa, Kishinev, 1971 (in Russian)

[8] Belousov V. D., $n$-Ary quasigroups, Shtiintsa, Kishinev, 1972 (in Russian) | MR

[9] Belousov V. D., Elements of Quasigroup Theory. A special course, Kishinev, 1981 (in Russian)

[10] Belousov V. D., Parastrophically orthogonal quasigroups, Shtiintsa, Kishinev, 1983 (in Russian) | MR

[11] Belousov V. D., Belyavskaya G. B., Latin squares, quasigroups and their applications, Shtiintsa, Kishinev, 1989 (in Russian) | MR

[12] Belyavskaya G. B., “Quasigroup power sets”, Quasigroups and Related Systems, 9 (2002), 1–19 | MR

[13] Bennett F., Hantao Zhang, “Latin Squares with Self-Orthogonal Conjugates”, Discrete Mathematics, 284:1–3 (2004), 45–55 | DOI | MR | Zbl

[14] Burris S., Sankappanavar H. P., A Course in Universal Algebra, Springer-Verlag, New York, 1981 | MR

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

[16] Damm M., Prüfziffersysteme über Quasigruppen, Diplomarbeit, Philipps-Universität Marburg, 1998

[17] Dénes J., Keedwell A. D., Latin Squares and their Applications, Académiai Kiadó, Budapest, 1974 | MR

[18] Dénes J., Keedwell A. D., Latin Squares. New Development in the Theory and Applications, Annals of Discrete Mathematics, 46, North-Holland, 1991 | MR | Zbl

[19] Dulmage A. L., Johnson D. M., Mendelsohn N. S., “Orthomorphisms of groups and orthogonal latin squares, I”, Canad. J. Math., 13 (1961), 356–372 | MR | Zbl

[20] Duplak J., “A parastrophic equivalence in quasigroups”, Quasigroups and Related Systems, 7 (2000), 7–14 | MR | Zbl

[21] Fraleigh J. B., A First Course in Abstract Algebra, Addison-Wesley, Reading, Massachusetts, 1982 | MR

[22] L. A. Skornyakov (ed.), General algebra, Nauka, Moscow, 1991 (in Russian) | MR | Zbl

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

[24] Keedwell A. D., Shcherbacov V. A., “Construction and properties of $(r,s,t)$-inverse quasigroups, II”, Discrete Math., 288 (2004), 61–71 | DOI | MR | Zbl

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

[26] Kishen K., “On the construction of latin and hyper-graceo-latin cubes and hypercubes”, J. Ind. Soc. Agric. Statist., 2 (1950), 20–48 | MR

[27] Laywine Ch. F., Mullen G. L., Discrete Mathematics Using Latin Squares, John Wiley Sons, Inc., New York, 1998 | MR

[28] Leakh I. V., On transformations of orthogonal systems of operations and algebraic nets, Ph. D. Dissertation, Institute of Mathematics, Kishinev, 1986, 108 pp., ages (in Russian)

[29] Lindner C. C., “Quasigroup identities and orthogonal arrays”, London Math. Soc., Lect. Note Ser., 82 (1983), 77–105 | MR | Zbl

[30] Mann H. B., “The construction of orthogonal latin squares”, Ann. Math. Statist., 13 (1942), 418–423 | DOI | MR | Zbl

[31] Mullen G. L., Shcherbacov V., “Properties of codes with one check symbol from a quasigroup point of view”, Buletinul Academiei de Ştiinţe a Republicii Moldova, Matematica, 2002, no. 3(40), 71–86 | MR | Zbl

[32] Mullen G. L., Shcherbacov V., “$n$-$T$-quasigroup codes with one check symbol and their error detection capabilities”, Comment. Math. Univ. Carolinae, 45:2 (2004), 321–340 | MR | Zbl

[33] Nĕmec P., Kepka T., “$T$-quasigroups, I”, Acta Universitatis, Carolinae Math. et Physica, 12:1 (1971), 39–49 | MR

[34] Norton D. A., “Group of orthogonal row-latin squares”, Pacific J. Math., 2 (1952), 335–341 | MR | Zbl

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

[36] Phelps K. T., “Conjugate orthogonal quasigroups”, J. Comb. Theory, A25:2 (1978), 117–127 | DOI | MR

[37] Rybnikov K. A., Introduction in combinatorial analysis, Publishing House of Moscow State University, Moscow, 1985 (in Russian) | MR

[38] Sade A., “Produit direct-singulier de quasigroupes othogonaux et anti-abeliens”, Ann. Soc. Sci. Bruxelles, Ser. I, 74 (1960), 91–99 | MR | Zbl

[39] Schöngardt E., “Über lateinische Quadrate und Unionen”, J. Reine Angew. Math., 163 (1930), 183–229

[40] Shcherbacov V. A., On automorphism groups and congruences of quasigroups, Thesis of Ph. D., IM AN MSSR, Kishinev, 1991, p. 88 (in Russian)

[41] Shcherbacov V. A., “About orthogonality of a quasigroup and its parastrophes”, International Conference on Radicals (ICOR-2003) (August 11–16, 2003, Chisinau, Moldova), 47–48

[42] Stein Sh. K., “On the foundations of quasigroups”, Trans. Amer. Math. Soc., 85:1 (1957), 228–256 | DOI | MR | Zbl