We investigate the congruence lattices of lattices in the varieties . Our approach is to represent congruences by open sets of suitable topological spaces. We introduce some special separation properties and show that for different n the lattices in have different congruence lattices.
We establish categorical dualities between varieties of isotropic median algebras and suitable categories of operational and relational topological structures. We follow a general duality theory of B.A. Davey and H. Werner. The duality results are used to describe free isotropic median algebras. If the number of free generators is less than five, the description is detailed.
We characterize d-lattices as those bounded lattices in which every maximal filter/ideal is prime, and we show that a d-lattice is complemented iff it is balanced iff all prime filters/ideals are maximal.
We say that a variety of algebras has the Compact Intersection Property (CIP), if the family of compact congruences of every is closed under intersection. We investigate the congruence lattices of algebras in locally finite, congruence-distributive CIP varieties and obtain a complete characterization for several types of such varieties. It turns out that our description only depends on subdirectly irreducible algebras in and embeddings between them. We believe that the strategy used here can...
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