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 research article, a new connection between serial fuzzy relations and an extended version of rough sets in algebraic structure quantale is established. The notion consists successor class overlap the element quantale. Thus approximation space based on via overlaps quantales, are introduced. main purpose study to provide basic structures fuzzy-relations. way, acquires certain appealing p...

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.

Two related results are proven in the modular invariant theory of GLn(Fq ). The first is a finite field analogue of a result of Springer on coinvariants of the symmetric group Sn acting on C[x1, . . . , xn]. It asserts that the following two Fqn [GLn(Fq )×F × q ]modules have the same composition factors: • the coinvariant algebra for GLn(Fq ) acting on Fqn [x1, . . . , xn], in which GLn(Fq ) ac...

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.

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.

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.