Localic sup-lattices and tropological systems

The approach to process semantics using quantales and modules is topologized by considering tropological systems whose sets of states are replaced by locales and which satisfy a suitable stability axiom. A corresponding notion of localic suplattice (algebra for the lower powerlocale monad) is described, and it is shown that there are contravariant functors from sup-lattices to localic sup-latices and, for each quantale Q, from left Q-modules to localic right Q-modules. A proof technique for third completeness due to Abramsky and Vickers is reset constructively, and an example of application to failures semantics is given. This is a preprint version of the article published as – Theoretical Computer Science 305 (2003), pp. 311–346. doi: 10.1016/S0304-3975(02)00702-8