On nondistributive Steiner quasigroups

• Autorzy: A. W. Marczak
• Języki publikacji: EN

Abstrakty

EN

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.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 1997
• Tom: 74
• Numer: 1
• Strony: 135-145

Daty

wydano

1997

otrzymano

1996-09-23

poprawiono

1996-12-13

Twórcy

autor

A. W. Marczak

• Institute of Mathematics, Technical University of Wrocław, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland

Bibliografia

• [1] S. Burris and R. Willard, Problem 17 of Grätzer and Kisielewicz, to appear.
• [2] J. Dénes and A. D. Keedwell, Latin Squares. New Developments to the Theory and Applications, Ann. Discrete Math. 46, North-Holland, Amsterdam, 1991.
• [3] J. Dudek, On the minimal extension of sequences, Algebra Universalis 23 (1986), 308-312.
• [4] J. Dudek, On Steiner quasigroups, Math. Slovaca 37 (1987), 71-83.
• [5] J. Dudek, The minimal extension of sequences II. On Problem 17 of Grätzer and Kisielewicz, Period. Math. Hungar., to appear.
• [6] J. Dudek, The minimal extension of sequences III. On Problem 16 of Grätzer and Kisielewicz, to appear.
• [7] J. Dudek and J. Gałuszka, A characterization of distributive Steiner quasigroups and semilattices, Discussiones Math. Algebra and Stochastic Methods 15 (1995), 101-119.
• [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.
• [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.
• [10] E. Marczewski, Independence and homomorphisms in abstract algebras, Fund. Math. 50 (1961), 45-61.
• [11] R. N. McKenzie, G. F. McNulty and W. A. Taylor, Algebras, Lattices, Varieties, Vol. 1, Wadsworth & Brooks/Cole, Monterey, Calif., 1987.