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 $$|A|\ge 4$$ .
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