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 of presheaves Set and the category of sheaves on L taking as covers of a the subsets C of L such that V C = a. This makes sense even if L is not complete, though we usually assume that L is a complete Heyting algebra. Both the category of presheaves and the category of sheaves are topoi, so they have a very nice logical structure and are not only cartesian closed but also locally cartesian closed. The logic which results is not, however, usually the kind encountered in fuzzy sets. It is intuitionistic. So the question in the title becomes: If L has another connective * which is sufficiently well behaved (like a t-norm on 10,11, for instance) what properties will be induced on the categories built on L. The presheaf and sheaf categories are not particularly good candidates for reflecting such a structure since their properties tend to come more from Sets than from the lattice properties, perhaps with sheafification needed. Looking at constructions like fuzzy sets and sets with a lattice valued equality seems more promising. In general, the categories we consider have either L-valued membership or L-valued equality. In either case there is an obvious structure for A ® B—take the product set A x B and take the * operation on the relevant membership or equality functions. We will need to show that the resulting structure has any other properties we put on our objects Fuzzy Sets and Systems 161 (2010) 1162-1174 Elsevier www.elsevier.com/locate/fss When Does a Category Built on a Lattice with a Monoidal Structure have a Monoidal Structure? L. Stout L.N. Stout / Fuzzy Sets and Systems 161 (2010) 1162–1174 1163 and that the maps needed are in fact maps in the relevant category. The isomorphisms in the definition of a monoidal category come from the isomorphisms for products in Sets, so the coherence conditions will be automatic. The question becomes: What properties do you need on to get nice properties on ⊗? 1. Conditions on the lattice of truth values We start with a lattice (L ,∧,∨) which can be considered to be a category using the order to give the morphisms. There are standard additional conditions one may place on L: 1. Existence of a top element . This is a common way to give a value for full truth. 2. Existence of a least element ⊥. This is a common way to give a value for full falsehood. 3. Completeness: existence of arbitrary suprema and infima, often needed to get a suitable quantification for first order logic. 4. Distributivity: For all a, b, c ∈ L we have a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c). 5. ∧∨ -distributivity: a ∧∨ ∈ b = ∨ ∈ (a ∧ b ). 6. ∨∧ -distributivity: a ∨∧ ∈ b = ∧ ∈ (a ∨ b ). Giving an alternate conjunction we usually ask for some of the following conditions: 1. Bifunctoriality: is a bifunctor from L × L to L, that is if a ≤ b and c ≤ d then a c ≤ b d. 2. Associativity: a (b c) = (a b) c. 3. Commutativity: a b = b a. 4. Tempered commutativity: a b c = a c b (my term for the key property used by Mulvey and Nawaz in [9]). 5. Idempotence: a a = a (which often forces a b = a ∧ b). 6. Nilpotence: for any a there is an n so that if we take n copies of a and combine them using we get a · · · a = ⊥ (for t-norms this is characteristic of the Łukaciewicz t-norm). 7. If L has a top element we can ask for (a) Right sided: a ≤ a with strict right sidedness if we get equality. (b) Left sided: a ≤ a with strict left sidedness if we get equality. 8. Units: a right unit u has a u = a for all a and a left unit has u a = a. If an operation has both right and left units they will be the same. They may not be . 9. If L has arbitrary ∨ we can ask for distributive laws: (a) Left distributive: a ∨ ∈ b = ∨ ∈ (a b ). (b) Right distributive: ( ∨ ∈ b ) a = ∨ ∈ (b a). 10. Autonomous or closed structures: (a) Left: a − : L → L has a right adjoint a ↘ − with a b ≤ c if and only if b ≤ a ↘ c. Note that this gives a (a ↘ c) ≤ c and tells us that a − will preserve any joins which exist since it is a left adjoint. (b) Right: − a : L → L has a right adjoint − ↙ a with b a ≤ c if and only if b ≤ c ↙ a. Note that this gives (c ↙ a) a ≤ c and tells us that − a will preserve any joins which exist. A monoidal structure on a partially ordered set L (thought of as a category) is given by a with bifunctoriality, associativity, and a two sided unit. It is a symmetric monoidal structure if it is commutative. It is monoidal closed if it satisfies autonomy. If L is a complete lattice and is distributive we have a simple formula for the closed structure: a ↘ b = ∨ {c|a c ≤ b} and b ↙ a = ∨ {c|c a ≤ b} A long literature gives many names for structures satisfying different collections of these axioms (ordered semigroups, lattice ordered semigroups, c.l.o.s.g’s, frames, quantales, Gelfand quantales, etc.). To avoid confusion I will specify which properties are being used in each construction. For Goguen’s category very little is needed; for categories based on an L-valued equality quite a lot more is needed. 1164 L.N. Stout / Fuzzy Sets and Systems 161 (2010) 1162–1174