Fuzzy semi-quantales, (L,M) quasi-fuzzy topological spaces and their duality

The present paper introduces M-fuzzy semi-quantales, fuzzifying semi-quantales, and (L, M)-quasi-fuzzy topological spaces, providing a common framework for (L, M)-fuzzy topological spaces of Kubiak and Šostak, L-quasi-fuzzy topological spaces of Rodabaugh and L-fuzzy topological spaces of Höhle and Šostak. In this paper, we set up a dual adjunction between the category of (L, M)-quasi-fuzzy topological spaces and the category of M-fuzzy semi-quantales, and then show that this adjunction includes a dual equivalence between the category of (L, M)-sober (L, M)-quasi-fuzzy topological spaces and the category of (L, M)-spatial M-fuzzy semi-quantales.