On calculating residuated approximations and the structure of finite lattices of small width

The concept of a residuated mapping relates to the concept of Galois connections; both arise in the theory of partially ordered sets. They have been applied in mathematical theories (e.g., category theory and formal concept analysis) and in theoretical computer science. The computation of residuated approximations between two lattices is influenced by lattice properties, e.g. distributivity.;In previous work, it has been proven that, for any mapping f : L → Q between two complete lattices L and Q , there exists a largest residuated mapping rhof dominated by f, and the notion of "the shadow sigma f of f" is introduced. A complete lattice Q is completely distributive if, and only if, the shadow of any mapping f : L → Q from any complete lattice L to Q is residuated.;Our objective herein is to study the characterization of the skeleton of a poset and to initiate the creation of a structure theory for finite lattices of small widths. We introduce the notion of the skeleton L˜ of a lattice L and apply it to find a more efficient algorithm to calculate the umbral number for any mapping from a ∼ finite lattice to a complete lattice.;We take a maximal autonomous chain containing x as an equivalent class [x] of x. The lattice L˜ is based on the sets {[x] | x ∈ L}. The umbral number for any mapping f : L → Q between two complete lattices is related to the property of L˜. Let L be a lattice satisfying the condition that [x] is finite for all x ∈ L; such an L is called ∼ finite. We define Lo = { &bigand; [x] | x ∈ L} and fo = f&vbm0;Lo . The umbral number for any isotone mapping f is equal to the umbral number for fo, and sa fo=sa f&vbm0;fo for any ordinal number alpha. Let uLo,Q be the maximal umbral number for all isotone mappings f : L → Q between two complete lattices. If L is a ∼ finite lattice, then uL,Q=uLo,Q . The computation of uLo,Q is less than or equal to that of uL,Q , we have a more efficient method to calculate the umbral number uL,Q .;The previous results indicate that the umbral number uL,Q determined by two lattices is determined by their structure, so we want to find out the structure of finite lattices of small widths. We completely determine the structure of lattices of width 2 and initiate a method to illuminate the structure of lattices of larger width.