Characteristic triangles of closure operators with applications in general algebra

Abstract

Our aim is to investigate groups and their weak congruence lattices in the abstract setting of lattices L with (local) closure operators C in the categorical sense, where L is regarded as a small category and C is a family of closure maps on the principal ideals of L. A useful tool for structural investigations of such lattices with closure is the so-called characteristic triangle, a certain substructure of the square L2. For example, a purely order-theoretical investigation of the characteristic triangle shows that the Dedekind groups (alias Hamiltonian groups) are precisely those with modular weak congruence lattices; similar results are obtained for other classes of algebras.