Danica Jakubíková-Studenovská – Overview on the lattice of congruence lattices of algebras on a finite set

P.J.Šafárik University, Košice, Slovakia
The subject of this talk is strongly related with the so called "concrete representation problem". Given a fixed finite set $A$, the congruence lattices of all algebras defined on $A$ ordered by inclusion form a finite lattice $\mathcal{E}_A$; the problem is to characterize those finite lattices $L$ which can be represented in the form $L\cong \mathcal{E}_A$ for some set $A$. This means that the properties of the lattice $\mathcal{E}_A$ has to be determined. We will present an overview of some known properties, and also of some open problems. Since the join- and meet-irreducible elements of a finite lattice are its "building-stones", an important objective is to describe these elements in $\mathcal{E}_A$. The join-irreducible elements of $\mathcal{E}_A$ have been completely characterized, while he problem on meet-irreducible elements of $\mathcal{E}_A$ has several, but only partial results.