Pointed semi-quantales and lattice-valued topological spaces

In a way more general than variable-basis approach to lattice-valued topological spaces, the present paper introduces an alternative approach to lattice-valued topological spaces-direct product representation spaces extending the notion of quantal spaces in the sense of Mulvey and Pelletier to semi-quantales recently proposed by Rodabaugh. This paper aims to give an answer to the main question whether there exists a categorical connection, possibly a categorical equivalence, between direct product representation spaces and variable-basis lattice-valued topological spaces. Small sources in the category of semi-quantales which are called pointed semi-quantales can be identified with direct product representation spaces. For this reason, the main problem will be handled in terms of pointed semi-quantales. Generalized quasi-lattice-valued topological spaces extending variable-basis quasi-topological spaces into the present setting are introduced to be a suitable topological counterpart of pointed semi-quantales. To formalize and to solve the main problem, categories of pointed semi-quantales and of generalized quasi-lattice-valued topological spaces are constructed, and the relations between these categories, providing a satisfactory answer to the main problem, are established in this paper.