On calculating residuated approximations

Abstract

For an order-preserving map f : L → Q between two complete lattices L and Q, there exists a largest residuated map ρ _f under f, which is called the residuated approximation of f. Andreka, Greechie, and Strecker introduced the notion of the shadow σ _f of f Iterations of the shadow are called the umbral mappings. The umbral mappings form a decreasing net that converges to the residuated approximation ρ _f of f. The umbral number u _f of f is the smallest ordinal number α such that the equation ${\sigma^{(\alpha)}_{f} = \rho_{f}}$ holds. In order to speed up the computation of the umbral number u _f of f and find some relation between the structure of L and u _f , we present the concept of the order skeleton of a lattice ${L, \tilde{L} = L/\sim}$, determined by a certain congruence relation ~ on L where each equivalence class [x] is the maximal autonomous chain containing x. If [x] is finite for each ${x \in L}$, then ${L_{o} := \{ \Lambda [x]\,|\, x \in L \}}$ is a join-subcomplete sub-semilattice of L isomorphic to the order skeleton ${\tilde{L}}$ of L; for every order-preserving mapping f : L → Q from such a lattice L to a complete lattice Q, we define f _o : L _o → Q by ${f_{o} := f|_{{L}_{o}}}$ and prove that ${u_{f} = u_{{f}_{o}}}$. For a lattice L with no infinite chains, the order skeleton ${\tilde{L}}$ of L is distributive if and only if the shadow σ _f of f is residuated for every complete lattice Q and every mapping f : L → Q. Related topics are discussed.