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Congruence relations on and varieties of directed multilattices. (English) Zbl 0642.06003
Several theorems, analogous to those concerning congruence relations of lattices, are proved for congruence relations of directed multilattices. For instance, congruence relations generated by sets of quotients in multilattices are characterized. With the aid of these results some questions concerning varieties of directed multilattices are investigated. First of all it is shown that the class of all modular directed multilattices as well as the class of all distributive ones are varieties. Moreover, if \(M_ 0,M_ 1,...,M_ n\) are finite directed, pairwise nonisomorphic multilattices that are not lattices, have only trivial congruence relations, and do not have proper subalgebras which are not lattices, then the variety generated by \(\{M_ 0,M_ 1,...,M_{n-1}\}\) is shown to be a proper subclass of the variety generated by \(\{M_ 0,M_ 1,...,M_ n\};\) consequently, in the lattice of varieties of directed multilattices there are infinite chains of distributive varieties. In contrast to the case of lattices, there are infinitely many varieties of distributive directed multilattices covering the variety of all distributive lattices in the lattice of varieties of directed multilattices.
Reviewer: G.Szász

MSC:
06B20 Varieties of lattices
06B10 Lattice ideals, congruence relations
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