Positivity relations on a locale

This paper analyses the notion of a positivity relationof Formal Topology from the point of view of the theory of Locales. It is shown that a positivity relation on a locale corresponds to a suitable class of points of its lower powerlocale. In particular, closed subtopologies associated to the positivity relation correspond to overt (that is, with open domain) weakly closed sublocales. Finally, some connection is revealed between positivity relations and localic suplattices (these are algebras for the powerlocale monad)

An algebraic generalization of Kripke structures

The Kripke semantics of classical propositional normal modal logic is made algebraic via an embedding of Kripke structures into the larger class of pointed stably supported quantales. This algebraic semantics subsumes the traditional algebraic semantics based on lattices with unary operators, and it suggests natural interpretations of modal logic, of possible interest in the applications, in structures that arise in geometry and analysis, such as foliated manifolds and operator algebras, via topological groupoids and inverse semigroups. We study completeness properties of the quantale based semantics for the systems K, T, K4, S4, and S5, in particular obtaining an axiomatization for S5 which does not use negation or the modal necessity operator. As additional examples we describe intuitionistic propositional modal logic, the logic of programs PDL, and the ramified temporal logic CTL.Comment: 39 page

Quantale Modules and their Operators, with Applications

The central topic of this work is the categories of modules over unital quantales. The main categorical properties are established and a special class of operators, called Q-module transforms, is defined. Such operators - that turn out to be precisely the homomorphisms between free objects in those categories - find concrete applications in two different branches of image processing, namely fuzzy image compression and mathematical morphology

Sheaves as modules

We revisit sheaves on locales by placing them in the context of the theory of quantale modules. The local homeomorphisms p : X → B are identified with the Hilbert B-modules that are equipped with a natural notion of basis. The homomorphisms of these modules are necessarily adjointable, and the resulting self dual category yields a description of the equivalence between local homeomorphisms and sheaves whereby morphisms of sheaves arise as the “operator adjoints” of the inverse images of the maps of local homeomorphisms.info:eu-repo/semantics/publishedVersio

Quantale Modules, with Applications to Logic and Image Processing

We propose a categorical and algebraic study of quantale modules. The results and constructions presented are also applied to abstract algebraic logic and to image processing tasks.Comment: 150 pages, 17 figures, 3 tables, Doctoral dissertation, Univ Salern