Category Of Distributive Lattices Research Articles

An algebraic study of exactness in partial contexts

DMFʼs are the natural algebraic tool for modelling reasoning with Körnerʼs partial predicates. We provide two representation theorems for DMFʼs which give rise to two adjunctions, the first between DMF and the category of sets and the second between DMF and the category of distributive lattices with minimum. Then we propose a logic L{1} for dealing with exactness in partial contexts, which belongs neither to the Leibniz, nor to the Frege hierarchies, and carry on its study with techniques of abstract algebraic logic. Finally a fully adequate and algebraizable Gentzen system for L{1} is given.

Spectra of rings and lattices

We construct a covariant functor from the category of distributive lattices with bottom and top whose morphisms are bottom and top preserving embeddings to the category of semisimple unital algebras over an arbitrary field whose morphisms are unital embeddings. The spectrum of a distributive lattice is homeomorphic to the spectrum of the ring (algebra) that is its image under this functor.

Stone?s representation theorem in fuzzy topology

In this paper, a complete solution to the problem of Stone's repesentation theorem in fuzzy topology is given for a class of completely distributive lattices. Precisely, it is proved that if L is a frame such that 0∈ L is a prime or 1∈ L is a coprime, then the category of distributive lattices is dually equivalent to the category of coherent L -locales and that if L is moreover completely distributive, then the category of distributive lattices is dually equivalent to the category of coherent stratified L -topological spaces.

A description of the projective Stone algebras

In his book [6] Grätzer sets a guideline for the research on Stone algebras; in view of his and Chen's triple characterization of Stone algebras [3], he considers a problem for these algebras solved if it can be reduced to a problem for Boolean algebras and one for distributive lattices with 1. In the same book he lists as Problem 53: describe the projective Stone algebras. For a Stone algebra L, let BL be its centre and DL its dense set. In order that L be projective, BL has to be a Boolean retract of BF for some free Stone algebra F, and DL has to be a retract of DF in the category of distributive lattices with 1. These conditions are, however, not sufficient, so one hopes to characterize the projective Stone algebras by adding further conditions, and in this spirit we in fact arrive at a description of these algebras. We also show, however, that every such algebra is a retract of some DF, F a free Stone algebra, so we conclude that there is no nice structural characterization of the projective Stone algebras along the line of Grätzer's programme.

A couple of triples

The standard contravariant adjunction between TOP (the category of topological spaces) and LAT (the category of distributive lattices) induces a triple Λ on LAT and a triple Σ on TOP. We show that the category LAT Λ of Λ-algebras is just the category of frames, and describe the category TOP Σ of Σ-algebras as a subcategory of TOP.

The semilattice tensor product of distributive lattices

We define the tensor product A ⊗ B A \otimes B for arbitrary semilattices A and B. The construction is analogous to one used in ring theory (see [4], [7], [8]) and different from one studied by A. Waterman [12], D. Mowat [9], and Z. Shmuely [10]. We show that the semilattice A ⊗ B A \otimes B is a distributive lattice whenever A and B are distributive lattices, and we investigate the relationship between the Stone space of A ⊗ B A \otimes B and the Stone spaces of the factors A and B. We conclude with some results concerning tensor products that are projective in the category of distributive lattices.

A Note on Stone Lattices

Stone lattices can be considered as forming a category of abstract algebras and thus there is a forgetful functor from this category to the category of distributive lattices with zero and unit. In this note we consider Stone lattices in this light (cf. [3], [4]) and describe an adjoint to the forgetful functor. The Stone extension of a distributive lattice with zero unit which we obtain differs markedly from the one given in [1].

Injective hulls in the category of distributive lattices.

Injective hulls in the category of distributive lattices.