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 Qmodule 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 fuzz...

This survey paper, specifically targeted at a readership of fuzzy logicians and fuzzy set theorists, aims to provide a gentle introduction to the basic notions of quantaloidenriched category theory. We discuss at length the definitions of quantaloid, quantaloidenriched category, distributor and functor, always giving several examples that – hopefully – appeal to the intended readership. To indi...

We use continuity spaces, a common refinement of posets and metric spaces, to develop a general theory of semantic domains which includes metric spaces and domains of cpo’s as special cases and provides the appropriate tools for producing new examples which may be suitable for modeling language constructs that occur in concurrent and probabilistic programming. Our proposal for a general notion ...

In this paper, we study the category of quantale-valued preordered spaces. We show that it is a normalized topological and give characterization zero-dimensionality D-connectedness in Moreover, characterize explicitly each T0, T1, pre-T2, T2 NT2 Finally, examine how these are related to other full subcategory Ti(pre-T2(L-Prord)) (i=0,1,2) pre-T2(L-Prord), Ti (L-Prord) (i=1,2) L-Prord isomorphic.

In a recent paper [7], J. W. Pelletier and J. Rosický published a characterization of *-simple *-quantales. Their results were adapted for the case of simple quantales by J. Paseka in [5]. In this paper we present similar characterizations which do not use a notion of discrete quantale. We also show a completely new characterization based on separating and cyclic sets. Further we explain a link...

A classical tensor product A⊗B of complete lattices A and B, consisting of all down-sets in A×B that are join-closed in either coordinate, is isomorphic to the complete lattice Gal(A,B) of Galois maps from A to B, turning arbitrary joins into meets. We introduce more general kinds of tensor products for closure spaces and for posets. They have the expected universal property for bimorphisms (se...

Linear logic is a logic of actions which seems well suited to various computer science applications. From its intrinsic ability to reeect computational resources, it is possible to reene diierent programming paradigms like formulae-as-types (proofs-as-programs) or formulae-as-states (proofs-as-computations) with a ner control on resource management. In the latter case, the correspondence betwee...

We introduce an algebraic approach to Dynamic Epistemic Logic. This approach has the advantage that: (i) its semantics is a transparent algebraic object with a minimal set of primitives from which most ingredients of Dynamic Epistemic Logic arise, (ii) it goes with the introduction of non-determinism, (iii) it naturally extends beyond boolean sets of propositions, up to intuitionistic and non-d...

For a quantale V, the category V-Top of V-valued topological spaces may be introduced as a full subcategory of those V-valued closure spaces whose closure operation preserves finite joins. In generalization of Barr’s characterization of topological spaces as the lax algebras of a lax extension of the ultrafilter monad from maps to relations of sets, for V completely distributive, V-topological ...