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 Hohle on quantales with properties making them commutative enough to have (non-symmetric) monoidal structures. If (L, A, v) is a lattice, then we can consider it to be a category using the order. Hence we can form both the category o