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The spectra of a variety of quasigroups and related combinatorial designs. (English) Zbl 0681.20042

The paper surveys some known and new results about finite quasigroups with given identities and their relations with conjugate orthogonal Latin squares, balanced and pairwise balanced designs and other related combinatorial structures.
Reviewer: V.Tonchev

MSC:

20N05 Loops, quasigroups
05B15 Orthogonal arrays, Latin squares, Room squares
05B30 Other designs, configurations
Full Text: DOI

References:

[1] Baker, R. D., Quasigroups and tactical systems, Aequationes Math., 18, 296-303 (1978) · Zbl 0393.05017
[2] Bennett, F. E., Latin squares with pairwise orthogonal conjugates, Discrete Math., 36, 117-137 (1981) · Zbl 0474.05017
[3] Bennett, F. E., On a class of \(n^2\) x 4 orthogonal arrays and associated quasigroups, Congressus Numerantium, 39, 117-122 (1983) · Zbl 0532.05010
[4] Bennett, F. E., On conjugate orthogonal idempotent Latin squares, Ars Combinatoria, 19, 37-50 (1985) · Zbl 0577.05015
[5] Bennett, F. E., Congugate orthogonal Latin squares and Mendelsohn designs, Ars Combinatoria, 19, 51-62 (1985) · Zbl 0577.05016
[6] Bennett, F. E., On \(r\)-fold perfect Mendelsohn designs, Ars Combinatoria, 23, 57-68 (1987) · Zbl 0645.05014
[7] Bennett, F. E., Pairwise balanced designs with prime power block sizes exceeding 7, Annals Discrete Math., 34, 43-64 (1987) · Zbl 0644.05012
[8] F.E. Bennett, Concerning pairwise balanced designs with prime power block sizes, Contemporary Math., to appear.; F.E. Bennett, Concerning pairwise balanced designs with prime power block sizes, Contemporary Math., to appear. · Zbl 0707.05010
[9] F.E. Bennett, Quasigroup identities and Mendelsohn designs, Canad. J. Math., to appear.; F.E. Bennett, Quasigroup identities and Mendelsohn designs, Canad. J. Math., to appear. · Zbl 0665.20035
[10] Bennett, F. E.; Mendelsohn, E.; Mendelsohn, N. S., Resolvable perfect cyclic designs, J. Combin. Theory, 29, A, 142-150 (1980) · Zbl 0444.05021
[11] Bennett, F. E.; Mendelsohn, N. S., On the spectrum of Stein quasigroups, Bull. Austral. Math. Soc., 21, 47-63 (1980) · Zbl 0416.20060
[12] Bennett, F. E.; Mendelsohn, N. S., Conjugate orthogonal Latin square graphs, Congressus Numerantium, 23, 179-192 (1979) · Zbl 0425.05012
[13] Bennett, F. E.; Wu, Lisheng; Zhu, L., Conjugate orthogonal Latin squares with equal-sized holes, Annals Discrete Math., 34, 65-80 (1987) · Zbl 0628.05011
[14] Bennett, F. E.; Wu, Lisheng; Zhu, L., On the existence of COLS with equal-sized holes, Ars Combinatoria, 26B, 5-36 (1988) · Zbl 0676.05021
[15] F.E. Bennett, Lisheng Wu and L. Zhu, Further results on incomplete (3,2,1)-conjugate orthogonal idempotent Latin squares, Discrete Math., to appear.; F.E. Bennett, Lisheng Wu and L. Zhu, Further results on incomplete (3,2,1)-conjugate orthogonal idempotent Latin squares, Discrete Math., to appear. · Zbl 0706.05013
[16] Beth, Th.; Jungnickel, D.; Lenz, H., Design Theory (1985), Bibliographisches Institut: Bibliographisches Institut Zurich · Zbl 0569.05002
[17] Brayton, R. K.; Coppersmith, D.; Hoffman, A. J., Self-orthogonal Latin squares of all orders \(n\)≠2, 3 or 6, Bull. Amer. Math. Soc., 80, 116-118 (1974) · Zbl 0277.05011
[18] Brouwer, A. E., Optimal packings of \(K_4\)’s into a \(K_n\), J. Combin. Theory, 26, A, 278-297 (1979) · Zbl 0412.05030
[19] Brouwer, A. E., The number of mutually orthogonal Latin squares - a table up to order 10000, (Research Report ZW123/79 (1979), Mathematisch Centrum: Mathematisch Centrum Amsterdam) · Zbl 0405.05013
[20] Brouwer, A. E., Four MOLS of order 10 with a hole of order 2, J. Statist. Planning and Inference, 10, 203-205 (1984) · Zbl 0553.05022
[21] Brouwer, A. E.; Hanani, H.; Schrijver, A., Group divisible designs with block-size four, Discrete Math., 20, 1-10 (1977) · Zbl 0371.62105
[22] Bruck, R. H., What is loop?, (Albert, A. A., Studies in Modern Algebra (1963), Prentice-Hall: Prentice-Hall Englewood Cliffs) · Zbl 0199.05202
[23] Colbourn, C. J.; Stinson, D. R., Edge-coloured designs with block size four, Research Report CORR 87-25 (1987) · Zbl 0658.05016
[24] Dénes, J.; Keedwell, A. D., Latin Squares and Their Applications (1974), Academic Press: Academic Press New York and London · Zbl 0283.05014
[25] Dinitz, J. H.; Stinson, D. R., MOLS with holes, Discrete Math., 44, 145-154 (1983) · Zbl 0507.05011
[26] Drake, D. A.; Larson, J. A., Pairwise balanced designs whose line sizes do not divide six, J. Combin. Theory, 34, A, 266-300 (1983) · Zbl 0518.05011
[27] Evans, T., Algebraic structures associated with Latin squares and orthogonal arrays, (Proc. Conf. Algebraic Aspects of Combinatorics, 13 (1975), Congressus Numerantium), 31-52 · Zbl 0318.05011
[28] Evans, T., Universal-algebraic aspects of combinatorics, (Proc. Internat. Conf. Universal Algebra. Proc. Internat. Conf. Universal Algebra, Janos Bolyai Math. Soc. (1980), North-Holland: North-Holland Amsterdam) · Zbl 0484.05014
[29] Evans, T., Finite representations of two-variable identities or Why are finite fields important in combinatorics?, Annals Discrete Math., 15, 135-141 (1982) · Zbl 0494.05010
[30] T. Evans, private communication.; T. Evans, private communication.
[31] Ganter, B., Combinatorial designs and algebras (1976), Technische Hochschule: Technische Hochschule Darmstadt, Preprint Nr. 270
[32] Ganter, B.; Werner, H., Equational classes of Steiner systems, Algebra Universalis, 5, 125-140 (1975) · Zbl 0312.08002
[33] Hanani, H., Balanced incomplete block designs and related designs, Discrete Math., 11, 255-369 (1975) · Zbl 0361.62067
[34] Hanani, H.; Ray-Chaudhuri, D. K.; Wilson, R. M., On resolvable designs, Discrete Math., 3, 343-357 (1972) · Zbl 0263.05016
[35] Horton, J. D., Sub-Latin squares and incomplete orthogonal arrays, J. Combin. Theory, 16, A, 23-33 (1974) · Zbl 0297.05015
[36] Hsu, D. F.; Keedwell, A. D., Generalized complete mappings, neofields, sequenceable groups and block designs. II, Pacific J. Math., 117, 291-312 (1985) · Zbl 0575.05011
[37] Keedwell, A. D., Circuit designs and Latin squares, Ars Combinatoria, 17, 79-90 (1984) · Zbl 0549.05013
[38] Lawless, J. F., Pairwise balanced designs and the construction of certain combinatorial systems, Proc. of the Second Louisiana Conf. on Combinatorics, Graph Theory, and Computing, 353-366 (1971), Baton Rouge · Zbl 0291.05009
[39] Lindner, C. C., Quasigroup identities and orthogonal arrays, (Lloyd, E. K., Surveys in Combinatorics. Surveys in Combinatorics, London Math. Soc. Lecture Notes Ser., 82 (1983), Cambridge Univ. Press), 77-105 · Zbl 0522.05014
[40] Lindner, C. C., Construction of quasigroups satisfying the identity \(x\)(xy)=yx, Canad. Math. Bull., 14, 57-59 (1971) · Zbl 0215.11502
[41] Lindner, C. C., Identities preserved by the singular direct product, Algebra Universalis, 1, 86-89 (1971) · Zbl 0221.20099
[42] Lindner, C. C., Identities preserved by the singular direct product II, Algebra Universalis, 2, 113-117 (1972) · Zbl 0249.20038
[43] Lindner, C. C., Construction of quasigroups using the singular direct product, Proc. Amer. Math. Soc., 29, 263-266 (1971) · Zbl 0217.08304
[44] C.C. Lindner, Identities preserved by group divisible designs, unpublished manuscript.; C.C. Lindner, Identities preserved by group divisible designs, unpublished manuscript.
[45] Lindner, C. C.; Mendelsohn, E., On the conjugates of an \(n^2\) x 4 orthogonal array, Discrete Math., 20, 123-132 (1977) · Zbl 0369.05010
[46] Lindner, C. C.; Mendelsohn, E.; Mendelsohn, N. S.; Wolk, B., Orthogonal Latin square graphs, J. Graph Theory, 3, 325-338 (1979) · Zbl 0422.05058
[47] Lindner, C. C.; Mendelsohn, N. S.; Sun, S. R., On the construction of Schroeder quasigroups, Discrete Math., 32, 271-280 (1980) · Zbl 0447.20050
[48] Lindner, C. C.; Mullin, R. C.; Hoffman, D. G., The spectra for the conjugate invariant subgroups of \(n^2\) x 4 orthogonal arrays, Canad. J. Math., 32, 1126-1139 (1980) · Zbl 0464.05012
[49] Lindner, C. C.; Steedly, D., On the number of conjugates of a quasigroup, Algebra Universalis, 5, 191-196 (1975) · Zbl 0324.20078
[50] MacNeish, H. F., Euler squares, Ann. Math., 23, 221-227 (1922) · JFM 48.0071.02
[51] Mendelsohn, N. S., Combinatorial designs as models of universal algebras, (Recent Progress in Combinatorics (1969), Academic Press: Academic Press New York and London), 123-132 · Zbl 0192.33302
[52] Mendelsohn, N. S., A natural generalization of Steiner triple systems, (Atkin, A. O.L.; Birch, B. J., Computers in Number Theory (1971), Academic Press: Academic Press New York), 323-338 · Zbl 0216.30102
[53] Mendelsohn, N. S., Perfect cyclic designs, Discrete Math., 20, 63-68 (1977) · Zbl 0369.05015
[54] Mendelsohn, N. S., Algebraic construction of combinatorial designs, Proc. Conf. Algebraic Aspects of Combinatorics. Proc. Conf. Algebraic Aspects of Combinatorics, Congressus Numerantium, 13, 157-168 (1975) · Zbl 0375.05016
[55] Mullin, R. C., A generalization of the singular direct product with application to skew Room squares, J. Combin. Theory, 29, A, 306-318 (1980) · Zbl 0461.05013
[56] Mullin, R. C.; Schellenberg, P. J.; Vanstone, S. A.; Wallis, W. D., On the existence of frames, Discrete Math., 37, 79-104 (1981) · Zbl 0465.05015
[57] R.C. Mullin and D.R. Stinson, Pairwise balanced designs with odd block sizes exceeding 5, Discrete Math., to appear.; R.C. Mullin and D.R. Stinson, Pairwise balanced designs with odd block sizes exceeding 5, Discrete Math., to appear. · Zbl 0703.05008
[58] Norton, D. A.; Stein, S. K., Cycles in algebraic systems, Proc. Amer. Math. Soc., 7, 999-1004 (1956) · Zbl 0077.02605
[59] Pelling, M. J.; Rogers, D. G., Stein quasigroups I: combinatorial aspects, Bull. Austral. Math. Soc., 18, 221-236 (1978) · Zbl 0377.20054
[60] Pelling, M. J.; Rogers, D. G., Stein quasigroups II: algebraic aspects, Bull. Austral. Math. Soc., 20, 321-334 (1979) · Zbl 0414.20059
[61] Phelps, K. T., Conjugate orthogonal quasigroups, J. Combin. Theory, 25, A, 117-127 (1978) · Zbl 0398.20086
[62] Ray-Chaudhuri, D. K.; Wilson, R. M., Solution of Kirkman’s schoolgirl problem, Amer. Math. Soc. Symp. Pure Math., 19, 187-203 (1971) · Zbl 0248.05009
[63] Sade, A., Produit direct-singulier de quasigroupes orthogonaux et anti-abéliens, Ann. Soc. Sci. Bruxelles Sér. I, 74, 91-99 (1960) · Zbl 0100.02204
[64] Seiden, E., A method of construction of resolvable BIBD, Sankhya, 25, A, 393-394 (1963) · Zbl 0124.10803
[65] Stein, S. K., On the foundations of quasigroups, Trans. Amer. Math. Soc., 85, 228-256 (1957) · Zbl 0079.02402
[66] Stein, S. K., Homogeneous quasigroups, Pacific J. Math., 14, 1091-1102 (1964) · Zbl 0132.26502
[67] Stinson, D. R., The equivalence of certain incomplete transversal designs and frames, Ars Combinatoria, 22, 81-87 (1986) · Zbl 0622.05012
[68] Stinson, D. R.; Zhu, L., On the existence of MOLS with equal-sized holes, Aequationes Math., 33, 96-105 (1987) · Zbl 0629.05017
[69] Todorov, D. T., Three mutually orthogonal Latin squares of order 14, Ars Combinatoria, 20, 45-48 (1985) · Zbl 0596.05009
[70] Wilson, R. M., Concerning the number of mutually orthgonal Latin squares, Discrete Math., 9, 181-198 (1974) · Zbl 0283.05009
[71] Wilson, R. M., Constructions and uses of pairwise balanced designs, Mathematical Centre Tracts, 55, 18-41 (1974) · Zbl 0312.05010
[72] Wilson, R. M., An existence theory for pairwise balanced designs I, J. Combin. Theory, 13, A, 220-245 (1972) · Zbl 0263.05014
[73] Wilson, R. M., An existence theory for pairwise balanced designs II, J. Combin. Theory, 13, A, 246-273 (1972) · Zbl 0263.05015
[74] Wilson, R. M., An existence theory for pairwise balanced designs III, J. Combin. Theory, 18, A, 71-79 (1975) · Zbl 0295.05002
[75] Zhang Xuebin, On the existence of \((v\),4,1)-PMD, Ars Combinatoria, to appear.; Zhang Xuebin, On the existence of \((v\),4,1)-PMD, Ars Combinatoria, to appear. · Zbl 0749.05016
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