From triangular scheme to Mal'tsev conditions
by
Eszter K. Horváth
University of Szeged, Bolyai Institute
Coauthors: Ivan Chajda, Gábor Czédli, Paolo Lipparini

From triangular scheme to Mal'cev conditions

Motivated by Gumm's Shifting Lemma, which asserts that congruence modular varieties satisfy a nice rectangular congruence scheme, Chajda investigated a triangular scheme, which is a consequence of congruence distributivity. Congruence distributive varieties satisfy this scheme not only for arbitrary three congruences but also for one tolerance and two congruences; i.e., the analogue of Gumm's Shifting Principle is valid. The investigations went on in different directions. First, the underlying reason for congruence schemes is that certain lattice indentities are equivalent with appropriate Horn sentences, called the shift of the lattice identity , however, not every lattice identity has a shift. Secondly, while the triangular scheme does not characterize congruence distributivity, an appropriate generalization called trapezoid scheme does. The third and probably the most important direction that grew out from the topic is the question if it is possible to put tolerances in place of all the three congruences. The answer is yes. As a special case, we obtain that in a congruence modular variety, R \cap S^* subset or equal (R \cap S)^* holds for any two tolerances R and S. As Radeleczki and Kearnes pointed out, this can easily be turned into a much more useful property, the so-called Tolerance Intersection Property, TIP for short, of congruence modular varieties: R^* \cap S^* = (R \cap S)^*. TIP has some applications. It is known that Tol L, the lattice of tolerances of a lattice L, has several nice properties discovered by Bandelt. Using TIP, these properties (some of them in a weaker form) can be extended to congruence distributive or congruence modular varieties, or varieties with a majority term. For example, if an algebra A has a majority term then Tol A is 0-modular, i.e., Tol A {0} contains no pentagon; the proof now is even simpler than Bandelt original one for lattices. Another application of TIP is about Mal'tsev conditions. Using TIP now we could prove that if p <= q is a lattice identitiy strong enough to imply modularity then p <= q has a Mal'tsev condition. This Mal'tsev condition is simply the conjunction of Day's condition and the Wille- Pixley's characterization of p_3 q. Here p_3 is the {, } term which we obtain from p by replacing joins by thre-fold relation product throughout. Where p <= q has previously known Mal'tsev condition then the Mal'tsev conditon extracted form p₃ subset or equal q is not as good as the known one, for it contains terms with too many variables. Much better Mal'tsev condition would come from p₂ subset or equal q instead of p₃ subset or equal q; the latest development is that this is possible.

Date received: January 30, 2003

Copyright © 2003 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 # cake-40.