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Keywords:
Quasigroup; idempotent groupoid term; mediality; incidence structure; parallelism; affine space; desarguesian affine plane
Quasigroup; idempotent groupoid term; mediality; incidence structure; parallelism; affine space; desarguesian affine plane
Summary:
Our aim is to demonstrate how the apparatus of groupoid terms (on two variables) might be employed for studying properties of parallelism in the so called $(n,k)$-quasigroups. We show that an incidence structure associated with a medial quasigroup of type $(n,k)$, $n>k\ge 3$, is either an affine space of dimension at least three, or a desarguesian plane. Conversely, if we start either with an affine space of order $k>2$ and dimension $m$, or with a desarguesian affine plane of order $k>2$ then there is a medial quasigroup of type $(k^m,k)$, $m>2$ such that the incidence structure naturally associated to a quasigroup is isomorphic with the starting one (the simplest case $k=2$ can be examined separately but is of little interest). The proofs are mostly based on properties of groupoid term functions, applied to idempotent medial quasigroups (idempotency means that $x\cdot x=x$ holds, and mediality means that the identity $(xy)(uv)=(xu)(yv)$ is satisfied).
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, (in Russian). MR 0218483
[3] Bruck, R. H.: A Survey of Binary Systems. Springer, Berlin, 1958. MR 0093552 | Zbl 0081.01704
[4] Denecke., K., Wismath, Sh. L.: Universal Algebra and Applications in Theoretical Computer Science. Chapman and Hall/CRC, 2002. MR 1887177 | Zbl 0993.08001
[5] Duplák, J.: On some permutations of a medial quasigroup. Mat. Čas. 24 (1974), 315–324, (in Russian). MR 0384971
[6] Duplák, J.: On some properties of transitive quasigroups. Zborník Ped. fak. Univ. Šafárika 1 (1976), 29–35, (in Slovak).
[7] Duplák, J.: Quasigroups and translation planes. J. Geom. 43 (1992), 95–107. DOI 10.1007/BF01245945 | MR 1148259
[8] Ganter, B., Werner, H.: Co-ordinatizing Steiner systems. Ann. Disc. Math. 7 (1980), 3–24. DOI 10.1016/S0167-5060(08)70167-5 | MR 0584400 | 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. MR 0972980 | 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. MR 0734873
[13] Kárteszi, F.: Introduction to Finite Geometries. Budapest, 1976. MR 0423175
[14] Lenz, H.: Über die Einführung einer absoluten Polarität in die projektive und affine Geometrie des Raumes. Math. Ann. 128 (1954), 363–373. DOI 10.1007/BF01360143 | MR 0067503 | Zbl 0056.13801
[15] Pflugfelder, H. O.: Quasigroups and Loops, Introduction. Heldermann Verlag, Berlin, 1990. MR 1125767 | Zbl 0715.20043
[16] Pukharev, N. K.: On $A^k_n$-algebras and finite regular planes. Sib. Mat. Zhur. 6, 4 (1965), 892–899, (in Russian).
[17] Pukharev, N. K.: On construction of $A^k_n$-algebras. Sib. Mat. Zhur. 7, 3 (1966), 724–727, (in Russian).
[18] Pukharev, N. K.: Geometric questions of some medial quasigroups. Sib. Mat. Zhur. 9, 4 (1968), 891–897, (in Russian). MR 0238170
[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, (in Russian). MR 0699124
[20] Romanowska, A., Smith, J. D. H.: Modal Theory, An Algebraic Approach to Order, Geometry, and Convexity. Heldermann Verlag, Berlin, 1985. MR 0788695 | Zbl 0553.08001
[21] Romanowska, A., Smith, J. D. H.: Modes. World Scientific, New Jersey, London, Singapore, Hong Kong, 2002. MR 1932199 | Zbl 1012.08001
[22] Szamkolowicz, L.: On the problem of existence of finite regular planes. Colloq. Math. 9 (1962), 245–250. MR 0142047 | Zbl 0106.14302
[23] Szamkolowicz, L.: Remarks on finite regular planes. Colloq. Math. 10 (1963), 31–37. MR 0164273 | Zbl 0118.15201
[24] Šiftar, J.: On affine planes over $A^k_n$-quasigroups. J. Geom. 20 (1983), 1–7. DOI 10.1007/BF01917989 | MR 0710059
[25] Stein, S. K.: Homogeneous quasigroups. Pacif. J. Math. 14 (1964), 1091–1102. DOI 10.2140/pjm.1964.14.1091 | MR 0170972 | Zbl 0132.26502
[26] Szmielew, W.: From Affine to Euclidean Geometry. Polish Scientific Publishers & D. Reidel Publishing Company, Warszawa & Dordrecht–Boston–London, 1983. MR 0720548 | Zbl 0516.51001
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