Luís Fernando Rodrigues de Sequeira
(CAUL, Universidade de Lisboa)
24/05/2002
Is modularity prime?
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). |
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