Representation of Distributive Lattices by means of ordered Stone Spaces

Abstract

Introduction Stone,in [8], developed for distributive lattices a representation theory generalizing that for Boolean algebras. This he achieved by topologizing the set X of prime ideals of a distributive lattice A (with a zero element) by taking as a base {P a : aeA} (where P a denotes the set of prime ideals of A not containing a), and by showing that the map a i-> P a is an isomorphism representing A as the lattice of all open compact subsets of its dual space X.The topological spaces which arise as duals of Boolean algebras may be characterized as those which are compact and totally disconnected (i.e. the Stone spaces); the corresponding purely topological characterization of the duals of distributive lattices obtained by Stone is less satisfactory. In the present paper we show that a much simpler characterization in terms of ordered topological spaces is possible. The representation theorem itself, and much of the duality theory consequent on it [8,6], becomes more natural in this new setting, and certain results not previously known can be obtained. It is hoped to give in a later paper a more detailed exposition of those aspects of the theory barely mentioned here.I should like to thank my supervisor, Dr. D. A. Edwards, for some helpful suggestions and also Dr. M. J. Canfell for permission to quote from his unpublished thesis.

Duality for distributive bisemilattices

“…Dualities for partially ordered structures are well known: there are Stone's dualities for Boolean algebras and distributive lattices [24,25], there is Priestley's duality for distributive lattices [18,19], and we have a Pontryagin duality for semilattices [8], to name just a few of them. All those dualities are constructed by the same method: one picks an object P that "lives" in both categories (that is, the category in question and its dual), and then utilizes the contravariant hom-functors Hom(-, P) in order to establish dualities between the two categories.…”

Bitopological duality for distributive lattices and Heyting algebras

“…Unlike Stone spaces, spectral spaces are not Hausdorff (not even T 1 ), and as a result, are more difficult to work with. In 1970 Priestley [6] described another dual category of DLat by means of special ordered Stone spaces, which became known as Priestley spaces, thus establishing that DLat is also dually equivalent to the category Pries of Priestley spaces and continuous order-preserving maps. Since DLat is dually equivalent to both Spec and Pries, it follows that the categories Spec and Pries are equivalent.…”

Isomorphism Universal Varieties of Heyting Algebras

“…The paper is subdivided into sections dealing respectively with elements of Priestley's duality applicable to varieties of Heyting algebras and their classification, with isomorphism universal varieties not generated by chains, the proof of isomorphism universality of the largest chain generated variety K, and of its smallest group universal subvariety K 4 . Throughout the paper, we use Priestley's duality for distributive lattices [16,17].…”