In this paper, we characterize explicitly the separation properties $T_0$ and $T_1$ at a point p in topological category of quantale-valued preordered spaces investigate how these characterizations are related. Moreover, prove that local hereditary productive.

Our work is a fundamental study of the notion of approximation inQ-categories and in (U,Q)-categories. Our exposition is categorical but kept as close to language of domain theory as possible. Consequently, we introduce auxiliary, approximating and Scott-continuous distributors, the way-below distributor, and continuity of Qand (U,Q)-categories. We fully characterize J-continuous Q-categories (...

Following the point-set lattice-theoretic (poslat) approach to topology of S. E. Rodabaugh [11], we introduced a more general way to do categorical (fuzzy) topology under the name of categorically-algebraic (catalg) topology [12], which unified many lattice-valued topological settings, and provided the means of intercommunication between different such frameworks. Moreover, motivated by the res...

A 2-form between two sup-lattices L and R is defined to be a suplattice bimorphism L×R → 2. Such 2-forms are equivalent to Galois connections, and we study them and their relation to quantales, involutive quantales and quantale modules. As examples we describe applications to C*-algebras.

We show that for a commutative quantale V every functor Set −→ V -cat has an enriched leftKan extension. As a consequence, coalgebras over Set are subsumed by coalgebras over V -cat. Moreover, one can build functors on V -cat by equipping Set-functors with a metric. 1998 ACM Subject Classification F.4.1 Mathematical Logic

Various notions of observing and testing processes are placed in a uniform algebraic framework in which observations are taken as constituting a quantale. General completeness criteria are stated, and proved in our applications. 2 67("/89:&",;&<+*91(8=&>)",#"3185&.781(2"#+.,"3&3.?+*&",;&0(.*188&81/",#+*8

in this work, we describe an adjunction between the comma category of set-based monads under the v -powerset monad and the category of associative lax extensions of set-based monads to the category of v -relations. in the process, we give a general construction of the kleisli extension of a monad to the category of v-relations.

A concept of quantum triad and its solution is introduced. It represents a common framework for several situations where we have a quantale with a right module and a left module, provided with a bilinear inner product. Examples include Van den Bossche quantaloids, quantum frames, simple and Galois quantales, operator algebras, or orthomodular lattices.