Optimal Mal'tsev conditions for modular congruence lattice identities
by
Gabor Czedli
University of Szeged, Bolyai Institute
Coauthors: Eszter K. Horvath, Paolo Lipparini

It was proved in [1] that in every congurence modular variety we have \alpha \cap \beta ^* subset or equal (\alpha \cap \beta) ^* for any tolerances \alpha and \beta. Here ^* stands for transitive closure. Based on [1], S. Radeleczki and K. Kearnes, independently, derived an even more useful property of tolerances in congruence modular varieties: \alpha ^* \cap \beta ^* =(\alpha \cap \beta) ^* . This property will be called tolerance intersection property , TIP in short.

Based on TIP, it was proved in [2] that for an arbitrary lattice identity implying modularity (or at least congruence modularity) there exists a Mal'tsev condition such that the identity holds in congruence lattices of algebras of a variety if and only if the variety satisfies the corresponding Mal'tsev condition. However, the Mal'tsev condition constructed in [2] is not the simplest known one in general. In [3] we improve this result by constructing the best Mal'tsev condition and various related conditions. In particular, if p and q are lattice terms and V is a congruence modular variety then the lattice identity p <= q holds for congruences of V iff p₂ subset or equal q holds for congruences of V where p₂ comes from p by substituting x o y for x \/ y everywhere.

As an application, Lipparini [3] gave a particularly easy new proof of Freese and Jónsson's result stating that modular congruence varieties are Arguesian, and he strengthened this result by replacing " Arguesian" by "higher Arguesian" in M. Haiman's sense.

Using TIP and commutator theory, Lipparini [3] showed that if p is an n-ary lattice term and \alpha₁, ... , \alpha_n are congruences in an arbitrary congruence modular variety then

p(\alpha1, ... , \alphan)=p(d)(\alpha1, ... , \alphan) o p2(\alpha1, ... , \alphan)

where p^(d) is the disjunctive normal form of p, computed as if we were in a distributive lattice. This result reminds us Gumm's famous paper "Congruence modularity is permutability composed with distributivity".

References (available at http://www.math.u-szeged.hu/ horvath/)

[1] G. Czédli and E. K. Horváth: Congruence distributivity and modularity permit tolerances, Acta Univ. Palacki. Olomuc., Fac. Rer. Nat., Mathematica, to appear.

[2] G. Czédli and E. K. Horváth: All congruence lattice identities implying modularity have Mal'tsev conditions, Algebra Universalis, to appear.

[3] G. Czédli, E. K. Horváth and P. Lipparini: Optimal Mal'tsev conditions in modular congruence varieties, Algebra Universalis, submitted.

Date received: June 11, 2002

Copyright © 2002 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caiv-23.