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...

Each divisible and unital quantale is associated with two quantaloids. Categories enriched over these two quantaloids can be regarded respectively as crisp sets endowed with fuzzy preorders and fuzzy sets endowed with fuzzy preorders. This paper deals with the relationship between these two kinds of enriched categories. This is a special case of the change-base issue in enriched category theory...

With the unit interval [0,1] as the truth value table, Chen and Wupresented the concept of  possibility computation over dcpos.Indeed, every possibility computation can be considered as a[0,1]-valued Scott open set on a dcpo. The aim of this paper is tostudy Chen-Wu's duality on quantale-valued setting. For clarity,with a commutative unital quantale $L$ as the truth value table, weintroduce a c...

We establish a close and previously unknown relation between quantales and groupoids, in terms of which the notion of étale groupoid is subsumed in a natural way by that of quantale. In particular, to each étale groupoid, either localic or topological, there is associated a unital involutive quantale. We obtain a bijective correspondence between localic étale groupoids and their quantales, whic...

motivated by the recent study on categorical properties of latticevalued topology, the paper considers a generalization of the notion of topological system introduced by s. vickers, providing an algebraic and a coalgebraic category of the new structures. as a result, the nature of the category   topsys   of s. vickers gets clari ed, and a metatheorem is stated, claiming that (latticevalu...

In this paper, the category of L-FINT is topological category over Set. We investigate the functorial relations between L-FTOP and LFINT where L is a strictly two-sided, commutative quantale lattice. Mathematics Subject Classification: 03E72, 54A40, 54B10

Suppose (Ω, ∗, I) is a commutative, unital quantale. Categories enriched over Ω can be studied as generalized, or many-valued, ordered structures. Because many concepts, such as complete distributivity, in lattice theory can be characterized by existence of certain adjunctions, they can be reformulated in the many-valued setting in terms of categorical postulations. So, it is possible, by aid o...

A structural semantics is developed for a first-order logic, with infinite disjunctions and conjunctions, that is characterised algebraically by quantales. The model structures involved combine the “covering systems” approach of Kripke-Joyal intuitionistic semantics from topos theory with the ordered groupoid structures used to model various connectives in substructural logics. The latter are u...

The ‘the’ in the title hides a subtlety. A metric space induces not one but four topologies - by means of open sets, closed closure, and interior they just so happen to coincide. agreement between these structures arising from function d:X×X→[0,∞] is due combination axioms lattice structure [0,∞]. Further motivation materializes Lawvere's observation 1973 effect that (slightly generalized) cate...