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.

Motivated by the concept of quantifier (in the sense of P. Halmos) on different algebraic structures (Boolean algebras, Heyting algebras, MV-algebras, orthomodular lattices, bounded distributive lattices) and the resulting notion of monadic algebra, the paper introduces the concept of a monadic quantale algebra, considers its properties and provides several representation theorems for the new s...

The notion of quantale, which designates a complete lattice equipped with an associative binary multiplication distributing over arbitrary joins, appears in various areasof mathematics-in quantaloid theory, non classical logic as completion the Lindebaum algebra, and different representations spectrum C∗ algebra asmany-valued commutative topologies. To put it briefly, its importance is nolonger...

1 Quantales In this section we introduce the basic deenitions and results of the theory of quantales (a good reference is Ros]). Quantales were introduced by Mulvey ((Mul]) as an algebraic tool for studying representations of non-commutative C-algebras. Informally, a quantale is a complete lattice Q equipped with a product distributive over arbitrary sup's. The importance of quantales for Linea...

For a quantale I, which is unit interval endowed with continuous triangular norm and the Barr extension β¯I of ultrafilter monad to I-Rel, characterization discrete presheaf associated given. It also proved that, when & Łucasiewicz norm, isomorphic saturated prefilter monad, product prime functional ideal submonad monad.

We know from a previous paper that the reticulation of coherent quantale A is bounded distributive lattice L ( ) whose prime spectrum homeomorphic to m -prime . This studies how can be used for transferring some properties lattices quantales and vice versa. shall illustrate this thesis by proving several results on pure w –pure elements means In particular, we investigate σ –ideals transferred ...

We introduce and present results about a class of quantales, the topological relational quantales, that can be associated with tuples (X,R) such that X is a topological space and R is a lower-semicontinuous equivalence relation on X (see Definition 1 below). Our motivating case studies and examples of topological relational quantales include concrete quantales such as the relational quantales o...

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.