Leírás
Abstract:
A variety in this talk is a class of models satisfying a set of identities. A variety V interprets in a variety W if a set of identities Z that defines V is satisfied by W if we replace the basic operation symbols in Z by some terms of W. Interpretation is a quasiorder relation on the class of varieties. By factoring out this quasiordered class with the corresponding equivalence, we obtain a lattice. The blocks of the quasiorder are called interpretability types, and the lattice is called the lattice of interpretability types of varieties. In their seminal paper Garcia and Taylor formulated the following two conjectures. The filter of the types of the congruence permutable varieties and the filter of the types of the congruence modular varieties are both prime filters in the lattice of interpretability types. Restricted to the idempotent varieties both conjectures were confirmed, but the general conjectures remained open, although more than twenty years ago Tschantz announced a still unpublished proof of the conjecture on congruence permutability. The present talk will be on a recent result we obtained for n-permutable varieties in a joint research with Gyenizse and Maróti. Note that the 2-permutable varieties are just the congruence permutable varieties. Our result states that for any n>4, the filter of the types of the n-permutable varieties is not prime in the lattice of interpretability types. The cases of n=2,3,4 remain open. As a consequence of the proof we obtain that the filter of interpretability types determined by the union of the n-permutable varieties for all n is not prime.
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