Topological Duality and Lattice Expansions, II: Lattice Expansions with Quasioperators

The main objective of this paper (the second of two parts) is to show that quasioperators can be dealt with smoothly in the topological duality established in Part I. A quasioperator is an operation on a lattice that either is join preserving and meet reversing in each argument or is meet preserving and join reversing in each argument. The paper discusses several common examples, including orthocomplementation on the closed subspaces of a fixed Hilbert space (sending meets to joins), modal operators auS and a- on a bounded modal lattice (preserving joins, resp. meets), residuation on a bounded residuated lattice (sending joins to meets in the first argument and meets to meets in the second). This paper introduces a refinement of the topological duality of Part I that makes explicit the topological distinction between the duals of meet homomorphisms and of join homomorphisms. As a result, quasioperators can be represented by certain continuous maps on the topological duals

Canonical extensions and ultraproducts of polarities

J{\'o}nsson and Tarski's notion of the perfect extension of a Boolean algebra with operators has evolved into an extensive theory of canonical extensions of lattice-based algebras. After reviewing this evolution we make two contributions. First it is shown that the failure of a variety of algebras to be closed under canonical extensions is witnessed by a particular one of its free algebras. The size of the set of generators of this algebra can be made a function of a collection of varieties and is a kind of Hanf number for canonical closure. Secondly we study the complete lattice of stable subsets of a polarity structure, and show that if a class of polarities is closed under ultraproducts, then its stable set lattices generate a variety that is closed under canonical extensions. This generalises an earlier result of the author about generation of canonically closed varieties of Boolean algebras with operators, which was in turn an abstraction of the result that a first-order definable class of Kripke frames determines a modal logic that is valid in its so-called canonical frames

Difference-restriction algebras of partial functions with operators: discrete duality and completion

We exhibit an adjunction between a category of abstract algebras of partial functions and a category of set quotients. The algebras are those atomic algebras representable as a collection of partial functions closed under relative complement and domain restriction; the morphisms are the complete homomorphisms. This generalises the discrete adjunction between the atomic Boolean algebras and the category of sets. We define the compatible completion of a representable algebra, and show that the monad induced by our adjunction yields the compatible completion of any atomic representable algebra. As a corollary, the adjunction restricts to a duality on the compatibly complete atomic representable algebras, generalising the discrete duality between complete atomic Boolean algebras and sets. We then extend these adjunction, duality, and completion results to representable algebras equipped with arbitrary additional completely additive and compatibility preserving operators.Comment: 30 pages. Small improvements throughou

Canonical extensions of locally compact frames

Canonical extension of finitary ordered structures such as lattices, posets, proximity lattices, etc., is a certain completion which entirely describes the topological dual of the ordered structure and it does so in a purely algebraic and choice-free way. We adapt the general algebraic technique that constructs them to the theory of frames. As a result, we show that every locally compact frame embeds into a completely distributive lattice by a construction which generalises, among others, the canonical extensions for distributive lattices and proximity lattices. This construction also provides a new description of a construction by Marcel Ern\'e. Moreover, canonical extensions of frames enable us to frame-theoretically represent monotone maps with respect to the specialisation order