On characterized varieties and quasivarieties of lattices
by
Anvar Nurakunov
Institute of Mathematics NAS, Bishkek, Kyrgyzstan
Coauthors: Victor Gorbunov (Institute of Mathematics SB RAS, Novosibirsk, Russsia)

The classical results in Lattice Theory by Dedekind [1] and Birkhoff [2] - a lattice is modular (distributive) if and only if it does not contain the pentagon N₅ (resp. N₅ and the 3-daimond M₃) as a sublattice - have been generalized by McKenzie [3] to notion of splitting algebras. We will consider varieties and quasivarieties defined by finite algebras not embeddable into algebras from those classes. It's easy to see that modular and distributive varieties are such varieties.

Let R, K are classes of algebras and R subset or equal K. A class of algebras R is K-characterized if there exist nonempty set M of finite algebras belonging to K such that R coincides with all algebras from K that do not contain algebras from M as subalgebras. If set M does not exist then we say that R is non K-characterized. If the set M isn't empty then algebra from the set M are called R-forbidden K-algebra. If set M is finite then R is called finitely K-characterized. If K is well-known class of algebras then we say that R is characterized (non-characterized, finitely characterized).

Further we will considerate lattices classes. But many definitions and results are true for any classes of algebras.

By R.Dedekind [1] (G.Birkhoff[2]) characterization we have that variety of modular lattices (variety of distributive lattices) is finitely characterized. Examples of non-characterized varieties are variety generated by all finite modular lattices and variety generated by all finite Desargues lattices. Nation's[4] counterexample to Jonsson's finite height conjecture give us possibility to construct non-characterized variety generated by finite lattice.

In [5] was constructed continuum non characterized varieties of lattices.We prove that there is continuum non characterized varieties of modular lattices and there is continuum non characterized locally finite varieties of lattices.

Other results are concerned to finite characterized quasivarieties, covers in lattice (quasi)varieties and R-forbidden lattices .

1. R.Dedekind , Uber die von drei Moduln erzeugte Dualgruppe, Math.Ann., 53(1900), 371-403.

2. G.Birkhoff, Abstract linear dependence and latticies, Amer.J.Math., 57(1935), 800-804.

3. R.McKenzie, Equational bases and nonmodular lattice varieties, Trans.Amer.Math.Soc., 174(1972), 1-43.

4. J.B.Nation, A counterexample to the finite height conjecture, Order 13(1996), 1-9.

5. V.A.Gorbunov, A.M.Nurakunov, J.A.Omarov, On number of non characterized varieties of lattices, (in Russian), 2-nd Int.Conf. on Math.Logic and Algebra, Barnaul(1991), Russia, Abstracts, vol.2, p.9.

Date received: January 8, 2002