Quantic Basis of Filter Theory

In this chapter we use commutative quantales as lattice–theoretic basis for the development of many valued filter theory. First we recall some basic definitions and facts of the theory of quantales (cf. [91]). A triple Q = (L, ≤, *) is called a commutative quantale iff (L, ≤) is a complete lattice and (L, *) is a commutative semigroup such that * is distributive over arbitrary joins in L. Since for every α ∈ L the map α * _ preserves arbitrary joins in L, it has a right adjoint denoted by α → _. Thus $$ \alpha * \gamma \leqslant \beta \Leftrightarrow \gamma \leqslant \alpha \to \beta $$