On nondistributive Steiner quasigroups
Marczak, A.
Colloquium Mathematicae, Tome 72 (1997), p. 135-145 / Harvested from The Polish Digital Mathematics Library
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A well known result of R. Dedekind states that a lattice is nonmodular if and only if it has a sublattice isomorphic to N5. Similarly a lattice is nondistributive if and only if it has a sublattice isomorphic to N5 or M3 (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.

Publié le : 1997-01-01
EUDML-ID : urn:eudml:doc:210496 
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     pages = {135-145},
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Marczak, A. On nondistributive Steiner quasigroups. Colloquium Mathematicae, Tome 72 (1997) pp. 135-145. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv74i1p135bwm/

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