Is Modularity prime?
by
Luís Sequeira
Universidade de Lisboa

The lattice L of interpretability types was defined by W. D. Neumann.

Garcia and Taylor conjectured that the filter of interpretability types of congruence-modular varieties is a prime filter of L. In contrast, it is well known that the filter of congruence-distributive varieties is not prime (it is even the intersection of strictly larger filters). Nonprimeness can be witnessed by building Jónsson terms of depth 2 in the join of two nondistributive varieties. There seems to be no case in the literature where terms of depth greater than 2 are required to show nonprimeness of a Maltsev filter.

The main result to be presented here is that terms of depth 2 alone cannot be used to disprove the conjecture. In other words, given two nonmodular varieties one cannot build Day terms of depth 2 for their join, in a similar manner to the ones known for Jónsson terms (in the join of some nondistributive varieties).

[Extended abstract can be found at http://www.lmc.fc.ul.pt/~lsequeir/math/extabstract.pdf ]

http://www.lmc.fc.ul.pt/~lsequeir/math/extabstract.pdf

Date received: January 8, 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 # caig-58.