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.

In fuzzy set theory non-idempotency arises when the conjunction is interpreted by arbitrary t-norms. There are many instances in mathematics where set theory ought to be non-commutative and/or non-idempotent. The purpose of this paper is to combine both ideas and to present a theory of non-commutative and non-idempotent quantale sets (among other things, standard concepts like fuzzy preorders a...

Taking advantage of the quantale-theoretic description \'etale groupoids we study principal bundles, Hilsum-Skandalis maps, and Morita equivalence in terms modules on inverse quantal frames. The Hilbert module quantale sheaves leads naturally to a formulation bimodules that resemble imprimitivity C*-algebras.

(Q,≦,∨) is a complete join semilattice (Q,・,e) is a monoid (∨S)・a=∨{b・a | b∈S} for a∈Q, S⊆Q a・(∨S)=∨{a・b | b∈S} for a∈Q, S⊆Q 2 Background: An Example of Quantale (Rel(A),≦,∨,・,e) is a quantale where Rel(A) is the set of all binary relations on A ≦ is the inclusion ∨ is the union ・ is the composition Q・R={ (a,c) | ∃b. (a,b)∈Q, (b,c)∈R } e is the identity relation 3 Our Goal: Relational Represent...

The topic of sets with fuzzy order relations valuated in complete lattices with additional structure has been quite active in the recent decade, and a number of papers have been published (see [3, 5, 6] among many others). Based on a quantale-valued order relation and subset membership, counterparts to common order-theoretic notions can be defined, like monotone mappings, adjunctions, joins and...

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...

In this paper we prove that any quantale Q is (isomorphic to) a quantale of suitable relations on Q. As a consequence two isomorphism theorems are also shown with suitable sets of functions of Q into Q. These theorems are the mathematical background one needs in order to give natural and complete semantics for (non-commutative) Linear Logic using relations. 1991 Mathematics Subject Classificati...

Our work is a foundational study of the notion of approximation inQ-categories and in (U,Q)-categories, for a quantale Q and the ultrafilter monad U. We introduce auxiliary, approximating and Scott-continuous distributors, the way-below distributor, and continuity of Qand (U,Q)-categories. We fully characterize continuous Q-categories (resp. (U,Q)-categories) among all cocomplete Q-categories (...

For an arbitrary localic étale groupoid G we provide simple descriptions, in terms of modules over the quantale O(G) of the groupoid, of the continuous actions of G, including actions on open maps and sheaves. The category of G-actions is isomorphic to a corresponding category of O(G)-modules, and as a corollary we obtain a new quantale based representation of étendues.