A new notion of a canonical extension A σ is introduced that applies to arbitrary bounded distributive lattice expansions (DLEs) A. The new definition agrees with the earlier ones whenever they apply. In particular, for a bounded distributive lattice A, A σ has the same meaning as before.A novel feature is the introduction of several topologies on the universe of the canonical extension of a DL. One of these topologies is used to define the canonical extension f σ : A σ → B σ of an arbitrary map f : A → B between DLs, and hence to define the canonical extension A σ of an arbitrary DLE A. Together the topologies form a powerful tool for showing that many properties of DLEs are preserved by canonical extensions.

The notion of a canonical extension of a lattice with additional operations is introduced. Both a concrete description and an abstract characterization of this extension are given. It is shown that this extension is functorial when applied to lattices whose additional operations are either order preserving or reversing, in each coordinate, and various results involving the preservation of identities under canonical extensions are established.

In this paper we consider distributive modal logic, a setting in which we may add modalities, such as classical types of modalities as well as weak forms of negation, to the fragment of classical propositional logic given by conjunction, disjunction, true, and false. For these logics we define both algebraic semantics, in the form of distributive modal algebras, and relational semantics, in the form of ordered Kripke structures. The main contributions of this paper lie in extending the notion of Sahlqvist axioms to our generalized setting and proving both a correspondence and a canonicity result for distributive modal logics axiomatized by Sahlqvist axioms. Our proof of the correspondence result relies on a reduction to the classical case, but our canonicity proof departs from the traditional style and uses the newly extended algebraic theory of canonical extensions.

A set of regular languages is a lattice of languages if and only if it can be defined by a set of profinite equations.The product on profinite words is the dual of the residuation operations on regular languages.In their more general form, our equations are of the form u → v, where u and v are profinite words. The first result not only subsumes Eilenberg-Reiterman's theory of varieties and their subsequent extensions, but it shows for instance that any class of regular languages defined by a fragment of logic closed under conjunctions and disjunctions (first order, monadic second order, temporal, etc.) admits an equational description. In particular, the celebrated McNaughtonSchützenberger characterisation of first order definable languages by the aperiodicity condition x ω = x ω+1 , far from being an isolated statement, now appears as an elegant instance of a very general result.How is this equational theory related to duality? The connection between profinite words and Stone spaces was already discovered by Almeida [2], [3, Theorem 3.6.1], but Pippenger [14] was the first to formulate it in terms of Stone duality. Almeida (implicitely) and Pippenger (explicitely) both observed that the Boolean algebra of regular languages over A * is dual to the Stone space A * , the set of profinite words. Pippenger actually came very close to our first result, since he mentioned that this duality extends to a one-to-one correspondence between Boolean algebras of regular languages and quotients of A * . Our first result is the full-fledged consequence of the similar one-to-one correspondence for all lattices of languages provided by Priestley duality.However, this link to duality theory is in fact much stronger and encompasses not only the underlying lattices and spaces involved but also the algebraic operations including the product of profinite words. That is the content of our

In this paper we introduce canonical extensions of partially ordered sets and monotone maps and a corresponding discrete duality. We then use these to give a uniform treatment of completeness of relational semantics for various substructural logics with implication as the residual(s) of fusion.