The Kernels of Skeletal Congruences on a Distributive Lattice

Abstract

AbstractLet L be a distributive lattice with 0 and C(L) be its lattice of congruences. The skeleton, SC(L), of C(L) consists of all those congruences which are the pseudocomplements of members of C(L), and is a complete BOOLEan lattice. An ideal is the kernel of a skeletal congruence if and only if it is an intersection of relative annihilator ideals, i.e. ideals of the form <r, s>j={x∈L: xΔr≤s} for suitable r, s∈L. The set KSC(L) of all such kernels forms an upper continuous distributive lattice and the map a ↦ (a={x∈L: x≤a} is a lower regular joindense embedding of L into KSC(L). The relationship between SC(L) and KSC(L) leads to numerous characterizations of disjunctive and generalized BOOLEan lattices. In particular, a distributive lattice L is disjunctive (generalized Boolean) if and only if the map Θ ↦ ker Θ is a lattice‐isomorphism of SC(L) onto KSC(L), whose inverse is the map J ↦ Θ (J)** (the map J ↦ Θ(J)). In addition, a study of KSC(L) leads to new simple proofs of results on the completions of special classes of lattices.