Geodesic
On nondistributive Steiner quasigroups
Colloquium Mathematicum, Tome 74 (1997) no. 1, pp. 135-145.

Voir la notice de l'article provenant de 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.
DOI : 10.4064/cm-74-1-135-145

A. Marczak 1

1 
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A. Marczak. On nondistributive Steiner quasigroups. Colloquium Mathematicum, Tome 74 (1997) no. 1, pp. 135-145. doi : 10.4064/cm-74-1-135-145. http://geodesic.mathdoc.fr/articles/10.4064/cm-74-1-135-145/

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