On lattices whose congruences form Stone lattices
A pseudocomplemented distributive lattice is called a Stone lattice if the pseudocomplement \(a^*\) of each element a has a complement. Several authors have studied lattices whose congruences form a Stone lattice. The present paper contains a general description of this class of lattices, and many of the earlier results follow as special cases of the current results. If \(a/b\to c/d\) denotes the fact that the quotient \(a/b\) is weakly projective into \(c/d\), then \(C(L)\) is a Stone lattice iff the pseudocomplement \(\theta^*\) of each congruence \(\theta\) is separable, and \(L\) satisfies: \(a/b\to c/d\) with \(a>b\) and \(c>d\) implies the existence of \(c', d'\) such that \(c\geq c'>d'\geq d\) and \(c'\equiv d'(\theta^{**}_{a/b})\).
scientific article; zbMATH DE number 4019123
scientific article