Duality as a Unifying Framework
by
Ingrid Rewitzky, Hilary Priestley, Mai Gehrke, Achim Jung, Marcello Bonsangue

DUALITY AS A UNIFYING FRAMEWORK A surprisingly uniform picture emerges when using duality for comparing various structures - notwithstanding their wide differences in formulation. With this workshop we bring together both known and new results pertaining to duality as a unifying framework.

Canonical Extensions Part I: Overview Hilary Priestley (Mathematics Institute, Oxford University) Formal languages of many different kinds nowadays play a central role in computer science and other disciplines. Semantic models of such languages, both algebraic and relational (with or without topology), are important in reasoning about them. In particular, modal logic, modelled algebraically by Boolean algebras with operators and purely relationally by Kripke frames, has a very well-developed theory, and deep and powerful theorems have been proved using correspondence techniques. Topological duality has been investigated for many logics which have algebraic models based on distributive lattices, for example intuitionistic logic and many-valued logics. Duality and correspondence were until quite recently researched separately. This is no longer so, and it is exciting to see a much bigger and much less fragmented picture emerging. The key to unification is the theory of canonical extensions, first studied in the context of BAOs, and now available in a far wider context. This provides algebraic versions, and generalizations, of some major theorems of modal logic, often with simpler and more transparent proofs. Significantly, too, canonical extensions are easy to work with beyond the setting of distributive lattices, where topological duality theory becomes much less tractable. Indeed, as Part II will indicate, the smoothest and nicest theory to date formulates the constructions for poset-ordered algebras! Here we have a blueprint for an algebraic version of duality and correspondence applicable to logics not covered by earlier approaches.

Part II: Latest developments Mai Gehrke (Department of Mathematical Sciences, New Mexico State University) We will start in the arena of distributive-lattice-ordered algebras. For canonicity to occur,it is necessary that the additional operations and the morphisms extend in a well-behaved way. We discuss the extension of maps in general. Here several topologies, including the interval topology as well as a stronger topology, special to canonical extensions, play a fundamental role: maps are extended using analogues of the well-known liminf and limsup constructions from real analysis. Semi-continuity properties of these extensions turn out to be the main tool underlying powerful theorems in the theory of canonical extensions. This leads to generalizations to the setting of general bounded lattice-ordered algebras of some major theorems from modal logic. Finally we will discuss very recent results on canonical extensions and correspondence for partially-ordered algebras that give a streamlined and uniform account of relational semantics for the implicative fragment of various substructural logics.

Duality theory as unifying framework for modal logics Marcello Bonsangue (LIACS, Leiden University) The aim of this talk is to describe a framework for the use of dualities for a coalgebraic semantics of modal logic. The approach is very general, and uses as main ingredient a duality between an algebraic category A and another category B. The framework gives a modal logic for each endofunctor L on A that is sound and complete with respect to a coalgebraic semantics induced by an endofunctor T on B dual to L. Moreover formulae are invariant under the behavioral equivalence induced by the coalgebra and the logic is expressive in the sense that non-bisimilar points are separated by some formula.

We apply the framework to Vietoris coalgebras on topological spaces, using the duality between spaces and observation frames, to obtain sound and complete modal logics for Posets, Sets, spectral spaces, Stone spaces and various categories of domains.

This is joint work with Alexander Kurz. A hierarchy of Stone dualities Achim Jung (Department of Computer Science, University of Birmingham) Stone's classical representation theorem (1936/37) associates with every Boolean algebra a compact totally disconnected space (now commonly referred to as a "Stone space"). The theory was extended to distributive lattices by Stone and, later, Priestley, and gave rise to ordered spaces which are still zero-dimensional.

From a topological point of view a different generalisation seems more desirable, namely, to general compact Hausdorff spaces. Somewhat surprisingly, this is indeed possible, as was shown recently by M. Andrew Moshier, building on earlier joint work with Mathias Kegelmann, Philipp Sunderhauf and the author. As in the classical case, the duality is between certain finitistic algebraic structures and topological spaces, and comes in an ordered and an unordered version.

While the duality is very pleasing at the object level, our understanding of the appropriate morphisms is less developed. More than one definition can be made and the talk will explore the relative merit of those choices.

This research was motivated by Domain Theory, in the sense of Scott's approach to Denotational Semantics of programming languages, and indeed many classes of domains are subsumed by the general construction. However, the primary structure of domains is order, not topology, and the challenge to capture this setting accurately in Stone duality is still open. In the talk I will present an approach towards its solution.

A duality for binary multirelations Ingrid Rewitzky (Department of Mathematics and Applied Mathematics, UCT) Specifications involving angelic and demonic nondeterminism may be thought of as a contract between two agents, both of which are free to make various choices. Such specifications have been represented syntactically in a monotone dynamic logic, algebraically by Boolean algebras with monotone operators, and relationally by binary multirelations. In this talk we show how duality provides a tight relationship between these representations, allowing translation between them. Finally, we extend this framework to a topological duality by defining a notion of multirelational Stone space.

Date received: May 31, 2004

Copyright © 2004 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts. Document # caop-07.