Congruence lattices in varieties with compact intersection property

summary:We say that a variety ${\mathcal V}$ of algebras has the Compact Intersection Property (CIP), if the family of compact congruences of every $A\in {\mathcal V}$ is closed under intersection. We investigate the congruence lattices of algebras in locally finite, congruence-distributive CIP varieties and obtain a complete characterization for several types of such varieties. It turns out that our description only depends on subdirectly irreducible algebras in ${\mathcal V}$ and embeddings between them. We believe that the strategy used here can be further developed and used to describe the congruence lattices for any (locally finite) congruence-distributive CIP variety