Topological representation of right-sided and idempotent quantales

Abstract

Prime elements of right-sided and idempotent quantales are identified with strong homomorphisms with values in the non-commutative, idempotent and right-sided chain with three elements. On this basis an adjunction between the category of right-sided idempotent quantales and the category of three-valued topological spaces is established. There is a second adjunction based on quantum spaces which is an extension of the duality between spatial right-sided idempotent quantales and sober quantum spaces. Both adjunctions restricts to the well known adjunction between topological spaces and locales. In particular, quantum spaces form a full subcategory of the category of three-valued topological spaces.