Congruences on Lattices and Lattice-Valued Functions

Abstract

For a complete lattice L and an L-valued function μ on a domain X, the cuts of μ determine a residuated map f from L to the power set of X ordered dually to inclusion. We describe a class of complete lattices for which the kernel of f is a complete congruence on L. Conversely, every complete congruence on a complete lattice L is uniquely determined by a suitable L-valued function μ on an arbitrary domain, as the kernel of a residuated map which sends every element p∈ L into the corresponding cut μp. As an application, using residuated maps we get a representation of finite lattices by meet-irreducible elements.