Mutual stochastic independences among-algebras and mutual algebraic inde-pendences among elements of semimodular lattices are observed to have a very similar behaviour. We suggest abstract independence structures called I-relations describing it. Presented examination of I-relations resembles a theory of abstract connectedness: a dual characterization of I-relations by families of connected set...

We prove that every finite distributive lattice D can be represented as the congruence lattice of a rectangular lattice K in which all congruences are principal. We verify this result in a stronger form as an extension theorem.

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