Quasigroup Identities and Mendelsohn Designs

Bennett, F. E.

Canadian journal of mathematics, Tome 41 (1989) no. 2, pp. 341-368

Résumé

A quasigroup is an ordered pair (Q, •), where Qis a set and (•) is a binary operation on Qsuch that the equations ax — b and ya — b are uniquely solvable for every pair of elements a,b in Q.It is well-known (see, for example, [11]) that the multiplication table of a quasigroup defines a Latinsquare, that is, a Latin square can be viewed as the multiplication table of a quasigroup with the headline and sideline removed. We are concerned mainly with finite quasigroups in this paper. A quasigroup (Q, •) is called idempotent if the identity x2 = x holds for all x in Q.

Détail

DOI : 10.4153/CJM-1989-017-0

Bibliographie

[1] 1. Bennett, F.E., The spectra of a variety of quasi groups and related combinatorial designs, Annals of Discrete Math., in press. Google Scholar

[2] 2. Bennett, F.E., On a class of n x 4 orthogonal arrays and associated quasigroups, Congressus Numerantium 39 (1983), 117–122. Google Scholar

[3] 3. Bennett, F.E., Conjugate orthogonal Latin squares and Mendelsohn designs, Ars Combinatoria 19 (1985), 51–62. Google Scholar

[4] 4. Bennett, F.E., On r-fold perfect Mendelsohn designs, Ars Combinatoria 23 (1987), 57–68. Google Scholar

[5] 5. Bennett, F.E., Mendelsohn, E. and Mendelsohn, N.S., Resolvable perfect cyclic designs, J. Combinatorial Theory (A) 29 (1980), 142–150. Google Scholar

[6] 6. Bennett, F.E., Lisheng Wu and Zhu, L., Conjugate orthogonal Latin squares with equal-sized holes, Annals of Discrete Math. 34 (1987), 65–80. Google Scholar

[7] 7. Bennett, F.E., On the existence of COLS with equal-sized holes, Ars Combinatoria to appear. Google Scholar

[8] 8. Th. Beth, Jungnickel, D. and Lenz, H., Design theory (Bibliographisches Institut, Zurich, 1985). Google Scholar

[9] 9. Brouwer, A.E., The number of mutually orthogonal Latin squares - a table up to order 10000, Research Report ZW123/79 (Mathematisch Centrum, Amsterdam, 1979). Google Scholar

[10] 10. Brouwer, A.E., Hanani, H. and Schrijver, A., Group divisible designs with block-size four, Discrete Math. 20 (1977), 1–10. Google Scholar

[11] 11. Dénes, J. and Keedwell, A.D., Latin squares and their applications (Academic Press, New York and London, 1974). Google Scholar

[12] 12. Dinitz, J.H. and Stinson, D.R., MOLS with holes, Discrete Math. 44 (1983), 145–154. Google Scholar

[13] 13. Evans, T., Algebraic structures associated with Latin squares and orthogonal arrays, Proc. Conf. Algebraic Aspects of Combinatorics, Congressus Numerantium 13 (1975), 31–52. Google Scholar

[14] 14. Evans, T., Universal-algebraic aspects of combinatorics, Proc. Internat. Conf. Universal Algebra, Janos Bolyai Math. Soc. (North-Holland, Amsterdam, 1980). Google Scholar

[15] 15. Evans, T., Finite representations of two-variable identities or Why are finite fields important in combinatories! Annals of Discrete math. 15 (1982), 135–141. Google Scholar

[16] 16. Ganter, B., Combinatorial designs and algebras, Preprint Nr. 270 (Technische Hochschule, Darmstadt, 1976). Google Scholar

[17] 17. Hanani, H., Balanced incomplete block designs and related designs. Discrete Math. 11 (1975), 255–369. Google Scholar

[18] 18. Hanani, H., Ray-Chaudhuri, D.K. and Wilson, R.M., On resolvable designs, Discrete Math. 3 (1972), 343–357. Google Scholar

[19] 19. Horton, J.D., Sub-Latin squares and incomplete orthogonal arrays, J. Combinatorial Theory (A) 16 (1974), 23–33. Google Scholar

[20] 20. Hsu, D.F. and Keedwell, A.D., Generalized complete mappings, neofields, sequenceable groups and block designs. 11, Pacific J. Math. 117 (1985), 291–312. Google Scholar

[21] 21. Keedwell, A.D., Circuit designs and Latin squares, Ars Combinatoria 17 (1984), 79–90. Google Scholar

[22] 22. Lindner, C.C., Quasigroup identities and orthogonal arrays, in: Surveys in combinatorics, London Math. Soc. Lecture Notes Ser. 82 (Cambridge Univ. Press, 1983), 77- 105. Google Scholar

[23] 23. Lindner, C.C., Construction of quasigroups satisfying the identity x(xy) = yx, Canad. Math. Bull. 14 (1971), 57–59. Google Scholar

[24] 24. Lindner, C.C., Identities preserved by the singular direct product, Algebra Universalis l ( 1971 ), 86–89. Google Scholar

[25] 25. Lindner, C.C., Identities preserved by the singular direct product II, Algebra Universalis 2 (1972), 113–117. Google Scholar

[26] 26. Lindner, C.C., Construction of quasigroups using the singular direct product, Proc. Amer. Math. Soc. 29(1971), 263–266. Google Scholar

[27] 27. Lindner, C.C., Identities preserved by group divisible designs, preprint. Google Scholar

[28] 28. Lindner, C.C., Mendelsohn, N.S. and Sun, S.R., On the construction of Schroeder quasigroups, Discrete Math. 32 (1980), 271–280. Google Scholar

[29] 29. Lindner, C.C. and Steedly, D., On the number of conjugates of a quasigroup, Algebra Universalis 5 (1975), 191–196. Google Scholar

[30] 30. MacNeish, H.F., Euler squares, Ann. Math. 23 (1922), 221–227. Google Scholar

[31] 31. Mendelsohn, N.S., Combinatorial designs as models of universal algebras in: Recent progress in combinatorics (Academic Press, New York and London, 1969), 123–132. Google Scholar

[32] 32. Mendelsohn, N.S., A natural generalization of Steiner triple systems, in: Computers in number theory (Academic Press, New York, 1971), 323–338. Google Scholar

[33] 33. Mendelsohn, N.S., Perfect cyclic designs, Discrete Math. 20 (1977), 63–68. Google Scholar

[34] 34. Mendelsohn, N.S., Algebraic construction of combinatorial designs, Proc. Conf. Algebraic Aspects of Combinatorics, Congressus Numerantium 13 (1975), 157–168. Google Scholar

[35] 35. Mullin, R.C., A generalization of the singular direct product with application to skew Room squares, J. Combinatorial Theory (A) 29 (1980), 306–318. Google Scholar

[36] 36. Mullin, R.C., Schellenberg, P.J., Vanstone, S.A. and Wallis, W.D., On the existence of frames, Discrete Math. 37 (1981), 79–104. Google Scholar

[37] 37. Mullin, R.C. and Stinson, D.R., Pairwise balanced designs with odd block sizes exceeding 5, Discrete Math., to appear. Google Scholar

[38] 38. Sade, A., Produit direct-singulier de quasigroupes orthogonaux et anti-abéliens, Ann. Soc. Sci. Bruxelles Sér. I 74 (1960), 91–99. Google Scholar

[39] 39. Seiden, E., A method of construction of resolvable BIBD, Sankhya (A) 25 (1963), 393–394. Google Scholar

[40] 40. Stein, S.K., On the foundations of quasigroups, Trans. Amer. Math. Soc. 85 (1957), 228–256. Google Scholar

[41] 41. Stein, S.K., Homogeneous quasigroups, Pacific J. Math. 14 (1964), 1091–1102. Google Scholar

[42] 42. Stinson, D.R. and Zhu, L., On the existence o/MOLS with equal-sized holes, Aequationes Math. 33 (1987), 96–105. Google Scholar

[43] 43. Todorov, D.T., Three mutually orthogonal Latin squares of order 14, Ars Combinatoria 20 (1985), 45-8. Google Scholar

[44] 44. Wilson, R.M., Concerning the number of mutually orthogonal Latin squares, Discrete Math. 9 (1974), 181–198. Google Scholar

[45] 45. Wilson, R.M., Constructions and uses of pairwise balanced designs, Mathematical Centre Tracts 55 (1974), 18-1. Google Scholar

[46] 46. Wilson, R.M., An existence theory for pairwise balanced designs I, J. Combinatorial Theory (A) 13 (1972), 220–245. Google Scholar

[47] 47. Wilson, R.M., An existence theory for pairwise balanced designs II, J. Combinatorial Theory (A) 13 (1972), 246–273. Google Scholar

[48] 48. Wilson, R.M., An existence theory for pairwise balanced designs III, J. Combinatorial Theory (A) 18 (1975), 71–79. Google Scholar