Equality in Linear Logic

1 Quantales In this section we introduce the basic deenitions and results of the theory of quantales (a good reference is Ros]). Quantales were introduced by Mulvey ((Mul]) as an algebraic tool for studying representations of non-commutative C-algebras. Informally, a quantale is a complete lattice Q equipped with a product distributive over arbitrary sup's. The importance of quantales for Linear Logic is revealed in Yetter's work ((Yet]), who proved that semantics of classical linear logic is given by a class of quantales, named Girard quantales, which coincides with Girard's phase semantics. An analogous result is obtained for a sort of non-commutative linear logic, as well as intuitionistic linear logic without negation, which suggest that the utilisation of the theory of quantales (or even weaker structures, such that *-autonomous posets) might be fruitful in studying the semantic of several variants of linear logic. As usual, we denote the order in a lattice by , while W and V denote the operations of sup and inf, respectively. We write > for the largest element in a lattice and 0 for its smallest element. Deenition 1.1 A quantale is a complete lattice Q with an associative binary operation : Q Q?! Q , which distributes on the right and on the left of arbitrary sup's, i.e.: Q1] a (b c) = (a b) c, for every a; b; c 2 Q Q2] a (W i2I a i) = W i2I (a a i), (W i2I a i) a = W i2I (a i a) A quantale Q is unital if it has an element 1 2 Q such that a1 = 1a = a, for every a 2 Q. A quantale Q is commutative if a b = b a, for every a, b 2 Q. A morphism of quantales is an operator between quantales which preserves and arbitrary sup's. It's easily seen that the above axioms imply that is increasing in both coordinates, that is If a b then, 8 c 2 Q, a c b c and c a c b We register a classic result: