The concept of a GADL as a generalization of an ADL is introduced. Necessary and sufficient conditions for a GADL to become a distributive lattice and a GADL to become an ADL are obtained. We also study the maximal sets in a GADL and give equivalent conditions for a GADL to become a distributive lattice in terms of maximal

It is well known that the only simple distributive lattice is the twoelement chain. In an earlier paper, we introduced the concept of a completesimple lattice, and proved the existence of infinite complete-simple distributive lattices. In this paper we provide a new proof which is easier to visualize.

To any entailment relation Sco74] we associate a distributive lattice. We use this to give a construction of the product of lattices over an arbitrary index set, of the Vietoris construction, of the embedding of a distributive lattice in a boolean algebra, and to give a logical description of some spaces associated to mathematical structures.

In this paper we describe a language and method for deriving ontologies and ordering databases. The ontological structures arrived at are distributive lattices with attribution operations that preserve ∨, ∧ and ⊥. The preservation of ∧ allows the attributes to model the natural join operation in databases. We start by introducing ontological frameworks and knowledge bases and define the notion ...

Let L be a distributive lattice and R(L) the associated Hibi ring. We compute regR(L) when L is a planar lattice and give bounds for regR(L) when L is nonplanar, in terms of the combinatorial data of L. As a consequence, we characterize the distributive lattices L for which the associated Hibi ring has a linear resolution.

Let L be a bounded distributive lattice. For k 1, let Sk (L) be the lattice of k-ary functions on L with the congruence substitution property (Boolean functions); let S(L) be the lattice of all Boolean functions. The lattices that can arise as Sk (L) or S(L) for some bounded distributive lattice L are characterized in terms of their Priestley spaces of prime ideals. For bounded distributive lat...

Let L be a lattice and let L1, L2 be sublattices of L. Let be a congruence relation of L1. We extend to L by taking the smallest congruence of L containing . Then we restrict to L2, obtaining the congruence L2 of L2. Thus we have de ned a map ConL1 ! ConL2. Obviously, this is an isotone 0-preservingmap of the nite distributive lattice ConL1 into the nite distributive lattice ConL2. The main res...