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 σ (α) f = ρ 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,L = L/∼, 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 ∈ L, then Lo :L} is a joinsubcomplete sub-semilattice of L isomorphic to the order skeletonL of L; for every order-preserving mapping f : L → Q from such a lattice L to a complete lattice Q, we define fo : Lo → Q by fo := f | Lo and prove that u f = u fo . For a lattice L with no infinite chains, the order skeletonL 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.
We introduce "π-versions" of five familiar conditions for distributivity by applying the various conditions to 3-element antichains only. We prove that they are inequivalent concepts, and characterize them via exclusion systems. A lattice L satisfies D 0π if a∧(b∨c) (a∧b)∨c for all 3-element antichains {a, b, c}. We consider a congruence relation ∼ whose blocks are the maximal autonomous chains and define the order-skeleton of a lattice L to be e L := L/∼. We prove that the following are equivalent for a lattice L: (i) L satisfies D 0π , (ii) e L satisfies any of the five π-versions of distributivity, (iii) the order-skeleton e L is distributive.
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