Compact Intersection Property and description of congruence lattices

Abstract

Abstract We say that a variety V of algebras has the Compact Intersection Property (CIP), if the family of compact congruences of every A ∈ V is closed under intersection. We investigate the congruence lattices of algebras in locally finite congruence-distributive CIP varieties. We prove some general results and obtain a complete characterization for some types of such varieties. We provide two kinds of description of congruence lattices: via direct limits and via Priestley duality.