For any ordered set P, the join dense completions of P form a complete lattice K(P) with least element O(P), the lattice of order ideals of P, and greatest element M(P), the Dedekind-MacNeille completion of P. The lattice K(P) is isomorphic to an ideal of the lattice of all closure operators on the lattice O(P). Thus it inherits some local structural properties which hold in the lattice of clos...

Given a reference lattice (X,⊑), we define fuzzy intervals to be the fuzzy sets such that their pcuts are crisp closed intervals of (X,⊑). We show that: given a complete lattice (X,⊑) the collection of its fuzzy intervals is a complete lattice. Furthermore we show that: if (X,⊑) is completely distributive then the lattice of its fuzzy intervals is distributive.

In 1990, we published the following result: Let m be a regular cardinal > א0. Every m-algebraic lattice L can be represented as the lattice of m-complete congruence relations of an m-complete modular lattice K. In this note, we present a short proof of this theorem. In fact, we present a significant improvement: The lattice K we construct is 2-distributive.

It is well known that closure operators on a complete lattice, ordered pointwise, give rise to a complete lattice, and this basic fact plays an important rôle in many fields of the semantics area, notably in domain theory and abstract interpretation. We strengthen that result by showing that closure operators on any directed-complete partial order (CPO) still form a complete lattice. An example...

The generalization of mathematical morphology to multivariate vector spaces is addressed in this paper. The proposed approach is fully unsupervised and consists in learning a complete lattice from an image as a nonlinear bijective mapping, interpreted in the form of a learned rank transformation together with an ordering of vectors. This unsupervised ordering of vectors relies on three steps: d...

This paper shows that any compactly generated lattice is a subdirect product of subdirectly irreducible lattices which are complete and upper continuous. An example of a compactly generated lattice which cannot be subdirectly decomposed into subdirectly irreducible compactly generated lattices is given. In the case of an ideal lattice of a lattice L, the decomposition into subdirectly irreducib...

In this work it is shown how fuzzy lattice neurocomputing (FLN) emerges as a connectionist paradigm in the framework of fuzzy lattices (FL¡framework) whose advantages include the capacity to deal rigorously with: disparate types of data such as numeric and linguistic data, intervals of values, \missing" and \don't care" data. A novel notation for the FL-framework is introduced here in order to ...

The theory of abelian totally ordered groups has a model completion. We show that the theory of abelian lattice-ordered groups has no model companion. Indeed, the Archimedean property can be captured by a first order V3V sentence for existentially complete abelian lattice-ordered groups, and distinguishes between finitely generic abelian lattice-ordered groups and infinitely generic ones. We th...