The aim of this paper is to extend the truth value table oflattice-valued convergence spaces to a more general case andthen to use it to introduce and study the quantale-valued fuzzy Scotttopology in fuzzy domain theory. Let $(L,*,varepsilon)$ be acommutative unital quantale and let $otimes$ be a binary operationon $L$ which is distributive over nonempty subsets. The quadruple$(L,*,otimes,varep...

-semilattices with that in the category of modules over a given unital commutative quantale. The resulting structures are called quantale algebroids. We show that their constitute a monadic category and prove a representation theorem for them using the notion of nucleus adjusted for our needs. We also characterize the lattice of nuclei on a free quantale algebroid. At the end of the paper, we p...

The notion of sobriety is extended to the realm topological spaces valued in a commutative and unital quantale, via an adjunction between category quantale modules quantale-valued spaces. Relations such sober domains based on flat ideals are investigated. In particular, it shown that if unit element coincides with its top element, then every domain (relative ideals) Scott topology space.

We associate a canonical unital involutive quantale to a topological groupoid. When the groupoid is also étale, this association is compatible with but independent from the theory of localic étale groupoids and their quantales [19] of P. Resende. As a motivating example, we describe the connection between the quantale and the C∗-algebra that both classify Penrose tilings, which was left as an o...

The construction of the formal ball model for metric spaces due to Edalat and Heckmann was generalized Q-categories by Kostanek Waszkiewicz, where Q is a commutative unital quantale. This paper concerns influence structure quantale on connection between Yoneda completeness directed their sets balls. In case that unit interval [0, 1] equipped with continuous t-norm &, it shown in order each Q-ca...

If the standard concepts of partial-order relation and subset are fuzzified, taking valuation in a unital commutative quantale Q, corresponding concepts of joins and join-preserving mappings can be introduced. We present constructions of limits, colimits and Hom-objects in categories Q-Sup of Q-valued fuzzy joinsemilattices, showing the analogy to the ordinary category Sup of join-semilattices.

For a monad S on a categoryK whose Kleisli category is a quantaloid, we introduce the notion of modularity, in such a way that morphisms in the Kleisli category may be regarded as V(bi)modules (= profunctors, distributors), for some quantale V . The assignment S // V is shown to belong to a global adjunction which, in the opposite direction, associates with every (commutative, unital) quantale ...

We introduce the concept of a Girard couple, which consists of two (not necessarily unital) quantales linked by a strong form of duality. The two basic examples of Girard couples arise in the study of endomorphism quantales and of the spectra of operator algebras. We construct, for an arbitrary sup-lattice S, a Girard quantale whose right-sided part is isomorphic to S.

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...

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...