The Structure of the Maximal Congruence Lattices of Algebras on a Finite Set
Danica Jakubíková-Studenovská, Reinhard Pöschel and Sándor Radeleczki
The congruence lattices of algebras with a fixed finite base set 𝐴 form a lattice Ε𝐴 (with respect to inclusion). The coatoms of Ε𝐴 are congruence lattices of monounary algebras (𝐴,𝑓), i.e., they are of the form Con(𝐴,𝑓) for a unary function 𝑓 : 𝐴 → 𝐴. It is known from  that there are three different types I, II, III of such coatoms which can be described explicitly by the corresponding type of 𝑓
In the present paper we are going to characterize these congruence lattices in detail. We prove that each coatom is a particular union of some nontrivial intervals of the partition lattice Eq(𝐴). Moreover, for each monounary algebra (𝐴,𝑓) of type I, II, III the join- and meet- irreducible elements, the atoms and the coatoms of its congruence lattice Con(𝐴, 𝑓) are determined, and the covering relation in this lattice is characterized.