A topological characterization of complete distributive lattices

AbstractAn ordered compact space is a compact topological space X, endowed with a partially ordered relation, whose graph is a closed set of X × X (cf. [4]). An important subclass of these spaces is that of Priestley spaces, characterized by the following property: for every x, y ϵ X with x ≰ y there is an increasing clopen set A (i.e. A is closed and open and such that a ϵ A, a ⩽ z implies that z ϵ A) which separates x from y, i.e., x ϵ A and y ≱ A. It is known (cf. [5, 6]) that there is a dual equivalence between the category Ld01 of distributive lattices with least and greatest element and the category P of Priestley spaces.In this paper we shall prove that a lattice L ϵ Ld01 is complete if and only if the associated Priestley space X verifies the condition: (E0) D ⊆ X, D is increasing and open implies D∗ is increasing clopen (where A∗ denotes the least increasing set which includes A).This result generalizes a well-known characterization of complete Boolean algebras in terms of associated Stone spaces (see [2, Ch. III, Section 4, Lemma 1], for instance).We shall also prove that an ordered compact space that fulfils (E0) is necessarily a Priestley space

Admissibility via Natural Dualities

It is shown that admissible clauses and quasi-identities of quasivarieties generated by a single finite algebra, or equivalently, the quasiequational and universal theories of their free algebras on countably infinitely many generators, may be characterized using natural dualities. In particular, axiomatizations are obtained for the admissible clauses and quasi-identities of bounded distributive lattices, Stone algebras, Kleene algebras and lattices, and De Morgan algebras and lattices.Comment: 22 pages; 3 figure

A Categorical View on Algebraic Lattices in Formal Concept Analysis

Formal concept analysis has grown from a new branch of the mathematical field of lattice theory to a widely recognized tool in Computer Science and elsewhere. In order to fully benefit from this theory, we believe that it can be enriched with notions such as approximation by computation or representability. The latter are commonly studied in denotational semantics and domain theory and captured most prominently by the notion of algebraicity, e.g. of lattices. In this paper, we explore the notion of algebraicity in formal concept analysis from a category-theoretical perspective. To this end, we build on the the notion of approximable concept with a suitable category and show that the latter is equivalent to the category of algebraic lattices. At the same time, the paper provides a relatively comprehensive account of the representation theory of algebraic lattices in the framework of Stone duality, relating well-known structures such as Scott information systems with further formalisms from logic, topology, domains and lattice theory.Comment: 36 page