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The lattice of congruence lattices of algebras on a finite set. (English) Zbl 1414.08001

The congruence lattices of all algebras defined on a fixed finite set \(A\) ordered by inclusion form a finite atomistic lattice \(E\). The authors describe the atoms, coatoms and irreducible elements of the lattice \(E\). It is proved that the lattice \(E\) is tolerance-simple, whenever the set \(A\) has at least four elements.

MSC:

08A30 Subalgebras, congruence relations
06B15 Representation theory of lattices
08A60 Unary algebras
06A15 Galois correspondences, closure operators (in relation to ordered sets)
08A35 Automorphisms and endomorphisms of algebraic structures
20M20 Semigroups of transformations, relations, partitions, etc.
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References:

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