Q-fuzzy subsets on ordered semigroups

In this paper, we introduce the concept of a Q-fuzzy subset of an ordered semigroup where a quantale Q replaces the unit interval. We define a binary operation on the fuzzy power set that makes it a quantale. In order to discuss the relation between the referential set and the fuzzy power set, the notion of an ordered Q-fuzzy point of an ordered semigroup is introduced. In terms of ordered Q-fuzzy points, we prove that any ordered semigroup can be embedded into a quantale, and also build a functor between the category of ordered semigroups and the category of quantales.