# On nondistributive Steiner quasigroups

Colloquium Mathematicae (1997)

- Volume: 74, Issue: 1, page 135-145
- ISSN: 0010-1354

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topMarczak, A.. "On nondistributive Steiner quasigroups." Colloquium Mathematicae 74.1 (1997): 135-145. <http://eudml.org/doc/210496>.

@article{Marczak1997,

abstract = {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.},

author = {Marczak, A.},

journal = {Colloquium Mathematicae},

keywords = {Steiner quasigroup; universal algebra; Steiner triple systems},

language = {eng},

number = {1},

pages = {135-145},

title = {On nondistributive Steiner quasigroups},

url = {http://eudml.org/doc/210496},

volume = {74},

year = {1997},

}

TY - JOUR

AU - Marczak, A.

TI - On nondistributive Steiner quasigroups

JO - Colloquium Mathematicae

PY - 1997

VL - 74

IS - 1

SP - 135

EP - 145

AB - 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.

LA - eng

KW - Steiner quasigroup; universal algebra; Steiner triple systems

UR - http://eudml.org/doc/210496

ER -

## References

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- [8] B. Ganter and H. Werner, Co-ordinatizing Steiner systems, in: Topics on Steiner Systems, C. C. Lindner and A. Rosa (eds.), Ann. Discrete Math. 7, North-Holland, Amsterdam, 1980, 3-24. Zbl0437.51007
- [9] G. Grätzer and A. Kisielewicz, A survey of some open problems on ${p}_{n}$-sequences and free spectra of algebras and varieties, in: Universal Algebra and Quasigroup Theory, A. Romanowska and J. D. H. Smith (eds.), Heldermann, Berlin, 1992, 57-88. Zbl0772.08001
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- [11] R. N. McKenzie, G. F. McNulty and W. A. Taylor, Algebras, Lattices, Varieties, Vol. 1, Wadsworth & Brooks/Cole, Monterey, Calif., 1987.

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