A note on atomistic weak congruence lattices (Q952678)

An element \(s\) of a lattice \(L\) is called distributive if \(s\vee(x \wedge y)=(s\vee x)\wedge(s\vee y)\), \(x,y\in L\). The dual identity defines codistributive elements. First, it is proved that a codistributive element \(s\) in an atomistic algebraic lattice \(L\) has a complement \(s'\) which is distributive; the kernels of the homomorphisms \(x\mapsto x \wedge s\), \(x\in L\), and \(x\mapsto x\vee s'\), \(x\in L\), coincide. Then some consequences to weak congruence lattices of algebras are derived. Namely, algebras with atomistic weak congruence lattices are studied in detail.