On Simple and Semisimple Quantales

In a recent paper [7], J. W. Pelletier and J. Rosický published a characterization of *-simple *-quantales. Their results were adapted for the case of simple quantales by J. Paseka in [5]. In this paper we present similar characterizations which do not use a notion of discrete quantale. We also show a completely new characterization based on separating and cyclic sets. Further we explain a link to simple quantale modules. To apply these characterizations, we study (*-)semisimple (*-)quantales and discuss some other perspectives. Our approach has connections with several earlier works on the subject [5, 7]. 1. Preliminaries Definition 1.1. By a sup-lattice is meant a complete lattice, a sup-lattice morphism is a mapping preserving arbitrary joins. If f : S → T is a sup-lattice morphism, the assignment f(s) ≤ t⇔ s ≤ f(t), explicitly f(t) = ∨ {s | f(s) ≤ t}, defines a mapping f : T → S preserving all meets. This mapping f is called the adjoint of f . The top element of a sup-lattice is denoted by 1, the bottom element by 0. Definition 1.2. By a quantale is meant a sup-lattice Q equipped with an associative multiplication which distributes over joins