interpretations, Stone duality, Curry-Howard isomorphism.

We establish a number of new results (and rederive some old results) concerning regular languages, using essentially topological methods. Our development is based on the duality (established by Stone) between Boolean algebras and certain topological spaces (which are now called \Stone spaces"). (This duality does not seem to have been recognized in the literature on regular languages, even thou...

It is a common trend in mathematics to study (natural) dualities for general algebraic structures and, in particular, for those arising from mathematical logic. The first step towards this direction traces back to the pioneering work by Stone for Boolean algebras [12]. Later on, Stone duality has been extended to the more general case of distributive lattices by Priestley [8], [9]. The two abov...

In this paper, we develop a duality for the varieties of a Lukasiewicz n + 1-valued modal system. This duality is an extension of Stone duality for modal algebras. Some logical consequences (such as completeness results, correspondence theory. . . ) are the derived and we propose some ideas for future research.

By abstract Stone duality we mean that the topology or contravariant powerset functor, seen as a self-adjoint exponential Σ on some category, is monadic. Using Beck’s theorem, this means that certain equalisers exist and carry the subspace topology. These subspaces are encoded by idempotents that play a role similar to that of nuclei in locale theory. Paré showed that any elementary topos has t...

In this paper we survey some recent developments in duality for lattices with additional operations paying special attention to Heyting algebras and the connections to Esakia’s work in this area. In the process we analyse the Heyting implication in the setting of canonical extensions both as a property of the lattice and as an additional operation. We describe Stone duality as derived from cano...

The problem of finding an algebraic structure for stable open subsets of a suitable domain has been recently raised by several authors. Specifically, it could be useful to have, in the stable case, a notion similar to the one of “frame”, in order to develop something similar to pointless topology, that is an algebraic insight of spaces. In domain theory, an interesting application of frame theo...

A brief outline of the topics of the course could be as follows. Coalgebras generalise transition systems. Modal logics are the natural logics for coalgebras. Stone duality provides the relationship between coalgebras and modal logic. Furthermore, some basic category theory is needed to understand coalgebras as well as Stone duality. So we will need to learn something about modal logic, about S...