×

Medial quasigroups of type \((n,k)\). (English) Zbl 1236.20066

The author demonstrates how the apparatus of groupoid terms might be employed for studying properties of parallelism in the (\(n,k\))-quasigroups. She considers a particular type of block designs admitting a high degree of regularity. The author pays attention to Steiner systems of type \(S(2,k,n)\) arising from idempotent medial quasigroups of a particular kind. She proves that each of them corresponds either to an affine space (\(n>k^2\), \(k>2\), in this case necessarily \(n=km\)) or to a Desarguesian affine plane (\(n=k^2\), \(k>3\)).

MSC:

20N05 Loops, quasigroups
05B25 Combinatorial aspects of finite geometries
PDF BibTeX XML Cite
Full Text: EuDML

References:

[1] Belousov, V. D.: Transitive distributive quasigroups. Ukr. Mat. Zhur 10, 1 (1958), 13-22.
[2] Belousov, V. D.: Foundations of the theory of quasigroups and loops. Nauka, Moscow, 1967)
[3] Bruck, R. H.: A Survey of Binary Systems. Springer, Berlin, 1958. · Zbl 0081.01704
[4] Denecke., K., Wismath, Sh. L.: Universal Algebra and Applications in Theoretical Computer Science. Chapman and Hall/CRC, 2002. · Zbl 0993.08001
[5] Duplák, J.: On some permutations of a medial quasigroup. Mat. Čas. 24 (1974), 315-324)
[6] Duplák, J.: On some properties of transitive quasigroups. Zborník Ped. fak. Univ. Šafárika 1 (1976), 29-35)
[7] Duplák, J.: Quasigroups and translation planes. J. Geom. 43 (1992), 95-107. · Zbl 0752.51001
[8] Ganter, B., Werner, H.: Co-ordinatizing Steiner systems. Ann. Disc. Math. 7 (1980), 3-24. · Zbl 0437.51007
[9] Havel, V. J., Vanžurová, A.: Medial Quasigroups and Geometry. Palacky University Press, Olomouc, 2006.
[10] Ihringer, Th.: Allgemeine Algebra. Teubner, Stuttgart, 1988. · Zbl 0661.08001
[11] Lindner, C. C., Rodger, C. A.: Design Theory. CRC Press, London, New York, Washington, 1997. · Zbl 0926.68090
[12] Ježek, J., Kepka, T.: Medial Groupoids. Academia, Praha, 1983. · Zbl 0538.08008
[13] Kárteszi, F.: Introduction to Finite Geometries. Budapest, 1976.
[14] Lenz, H.: Über die Einführung einer absoluten Polarität in die projektive und affine Geometrie des Raumes. Math. Ann. 128 (1954), 363-373. · Zbl 0056.13801
[15] Pflugfelder, H. O.: Quasigroups and Loops, Introduction. Heldermann Verlag, Berlin, 1990. · Zbl 0715.20043
[16] Pukharev, N. K.: On \(A^k_n\)-algebras and finite regular planes. Sib. Mat. Zhur. 6, 4 (1965), 892-899)
[17] Pukharev, N. K.: On construction of \(A^k_n\)-algebras. Sib. Mat. Zhur. 7, 3 (1966), 724-727)
[18] Pukharev, N. K.: Geometric questions of some medial quasigroups. Sib. Mat. Zhur. 9, 4 (1968), 891-897) · Zbl 0212.25501
[19] Pukharev, N. K.: Some properties of groupoids and quasigroups connected with balanced incomplete block schemes. Quasigroups and Latine squares, Mat. Issl., Kishinev 71 (1983), 77-85) · Zbl 0551.20052
[20] Romanowska, A., Smith, J. D. H.: Modal Theory, An Algebraic Approach to Order, Geometry, and Convexity. Heldermann Verlag, Berlin, 1985. · Zbl 0553.08001
[21] Romanowska, A., Smith, J. D. H.: Modes. World Scientific, New Jersey, London, Singapore, Hong Kong, 2002. · Zbl 1012.08001
[22] Szamkolowicz, L.: On the problem of existence of finite regular planes. Colloq. Math. 9 (1962), 245-250. · Zbl 0106.14302
[23] Szamkolowicz, L.: Remarks on finite regular planes. Colloq. Math. 10 (1963), 31-37. · Zbl 0118.15201
[24] Šiftar, J.: On affine planes over \(A^k_n\)-quasigroups. J. Geom. 20 (1983), 1-7. · Zbl 0514.51010
[25] Stein, S. K.: Homogeneous quasigroups. Pacif. J. Math. 14 (1964), 1091-1102. · Zbl 0132.26502
[26] Szmielew, W.: From Affine to Euclidean Geometry. Polish Scientific Publishers & D. Reidel Publishing Company, Warszawa & Dordrecht-Boston-London, 1983. · Zbl 0516.51001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.