The lattice of congruence lattices of algebras on a finite set
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Abstract
The congruence lattices of all algebras defined on a fixed finite set A ordered by inclusion form a finite atomistic lattice \(\mathcal {E}\). We describe the atoms and coatoms. Each meet-irreducible element of \(\mathcal {E}\) being determined by a single unary mapping on A, we characterize completely those which are determined by a permutation or by an acyclic mapping on the set A. Using these characterisations we deduce several properties of the lattice \(\mathcal {E}\); in particular, we prove that \(\mathcal {E}\) is tolerance-simple whenever \(|A|\ge 4\).
Keywords
Congruence lattice Unary operation Monounary algebra Join-irreducible element Meet-irreducible element Tolerance simpleMathematics Subject Classification
Primary 08A30 Secondary 06B15 08A60 06A15 08A35 08A99 20M20References
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