Publisher Summary This chapter introduces the basic notions in theory of quantales. It describes the aspects of algebraic and categorical properties of quantales and quantale modules. This chapter… Expand

Using this approach to V-semilattices with a duality i.e. Hilbert 2-modules, a nuclear traced ideal is described and nuclear maps in the respective category are characterized.Expand

Every quantale can be embedded into a simple quantale and every
involutive quantale can be embedded into a somple involutive
quantale. The image satisfies an analog of the von Neumann
bicommutant… Expand

Quantales can be viewed as a framework for a non—commutative topology. Basic properties of quantales are given. There are considered regular and normal quantales. The main, result asserts that… Expand

In this paper, we continue the study of quantum B-algebras with emphasis on filters on integral quantum B-algebras. We then study filters in the setting of pseudo-hoops. First, we establish an… Expand

For compactly generated Archimedean lattice effect algebras the equivalence of (o)-continuity of states with their complete additivity is proved and the “State smearing theorem” is proved.Expand

The interior tensor product in the context of the theory of
Hilbert modules over involutive quantales is studied.
Associativity, functoriality and representability via compact
maps of the interior… Expand

The injectives in the category of S-semigroups with subhomomorphisms generalize the quantale algebras introduced by Solovyov, which provide a convenient universally algebraic framework for developing lattice-valued analogues of fuzzification.Expand