On nondistributive Steiner quasigroups
Colloquium Mathematicum, Tome 74 (1997) no. 1, pp. 135-145
Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences
A well known result of R. Dedekind states that a lattice is nonmodular if and only if it has a sublattice isomorphic to $N_5$. Similarly a lattice is nondistributive if and only if it has a sublattice isomorphic to $N_5$ or $M_3$ (see [11]). Recently a few results in this spirit were obtained involving the number of polynomials of an algebra (see e.g. [1], [3], [5], [6]). In this paper we prove that a nondistributive Steiner quasigroup (G,·) has at least 21 essentially ternary polynomials (which improves the recent result obtained in [7]) and this bound is achieved if and only if (G,·) satisfies the identity (xz·yz)·(xy)z = (xz)y·x. Moreover we prove that a Steiner quasigroup (G,·) with 21 essentially ternary polynomials contains isomorphically a certain Steiner quasigroup (M,·), which we describe in Section 1.
@article{10_4064_cm_74_1_135_145,
author = {A. Marczak},
title = {On nondistributive {Steiner} quasigroups},
journal = {Colloquium Mathematicum},
pages = {135--145},
year = {1997},
volume = {74},
number = {1},
doi = {10.4064/cm-74-1-135-145},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/cm-74-1-135-145/}
}
A. Marczak. On nondistributive Steiner quasigroups. Colloquium Mathematicum, Tome 74 (1997) no. 1, pp. 135-145. doi: 10.4064/cm-74-1-135-145
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