Standard \(n\)-ideals of a lattice (Q1377379)

A convex sublattice of a lattice \(L\) containing a fixed element \(n\) of \(L\) is called an \(n\)-ideal of \(L\). An \(n\)-ideal \(S\) of \(L\) is called a standard \(n\)-ideal if \(S\) is a standard element in the lattice of all \(n\)-ideals of \(L\). The authors show that for a neutral element \(n\) of a lattice \(L\), an \(n\)-ideal is standard iff it is a standard sublattice. Furthermore a description of the smallest congruence relation containing a standard \(n\)-ideal as a class is given.