Congruence lattices of \(p\)-algebras
A \(p\)-algebra is a bounded lattice endowed with a unary operation \({}^*\) satisfying \(x\leq a^*\) iff \(x\wedge a=0\). The authors give a new proof of the following theorem due to Katriňák: Every finite distributive lattice is isomorphic to the congruence lattice of a finite \(p\)-algebra. Then they indicate how to modify the proof so as to obtain the following generalization. If \(D\) is an algebraic distributive lattice such that the unit element of \(D\) is compact and every compact element of \(D\) is a finite join of join-irreducible compact elements, then \(D\) can be represented as the congruence lattice of a \(p\)-algebra. Finally some open problems are stated.
scientific article; zbMATH DE number 780793