Quantale algebras as a generalization of lattice-valued frames

Recently, I. Stubbe constructed an isomorphism between the categories of right Q-modules and cocomplete skeletal Q-categories for a given unital quantale Q. Employing his results, we obtain an isomorphism between the categories of Q-algebras and Q-quantales, where Q is additionally assumed to be commutative. As a consequence, we provide a common framework for two concepts of lattice-valued frame, which are currently available in the literature. Moreover, we obtain a convenient setting for lattice-valued extensions of the famous equivalence between the categories of sober topological spaces and spatial locales, as well as for answering the question on its relationships to the notion of stratification of lattice-valued topological spaces.