When does a category built on a lattice with a monoidal structure have a monoidal structure?

In a word, sometimes. And it gets harder if the structure on L is not commutative. In this paper we consider the question of what properties are needed on the lattice L equipped with an operation ⋆ for several different kinds of categories built using Sets and L to have monoidal and monoidal closed structures. This works best for the Goguen category Set ( L ) in which membership, but not equality, is made fuzzy and maps respect membership. Commutativity becomes critical if we make the equality fuzzy as well. This can be done several ways, so a progression of categories is considered. Using sets with an L-valued equality and functions which respect that equality gives a monoidal category which is closed if we use a strong form of the transitive law. If we use strict extensional total relations and a strong transitive law (and ⋆ is commutative and nearly idempotent), we get a monoidal structure. We also recall some constructions by Mulvey, Nawaz, and Höhle on quantales with properties making them commutative enough to have (non-symmetric) monoidal structures.