On nondistributive Steiner quasigroups

A. Marczak

Colloquium Mathematicae (1997)

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

Abstract

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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 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.

How to cite

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Marczak, 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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  7. [7] J. Dudek and J. Gałuszka, A characterization of distributive Steiner quasigroups and semilattices, Discussiones Math. Algebra and Stochastic Methods 15 (1995), 101-119. Zbl0835.08004
  8. [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. [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
  10. [10] E. Marczewski, Independence and homomorphisms in abstract algebras, Fund. Math. 50 (1961), 45-61. Zbl0104.25501
  11. [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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