A representation theorem for quantale algebras ∗

While the study of quantale-like structures goes back up to the 1930’s (not-withstanding that the term itself was introduced in [9] in connection with certain aspects of C∗-algebras), there has recently been much interest in quantales in a variety of contexts. The most important connection probably is with Girard’s linear logic (see, e.g., [3]). In particular, [1] enunciates the following slogan: Quantales are to linear logic as frames are to the intuitionistic one. With this paper we hope to contribute to the theory of quantales. It considers the notion of quantale algebra and shows a representation theorem for such structures generalizing the following representation theorem for quantales from [13] (notice that P(−) stands for the power-set): Theorem 1. If Q is a quantale, there is a semigroup S and a quantic nucleus P(S) j− → P(S) such that Q ∼ = P(S)j. To be precise we start by introducing the category Q-Alg of algebras over a given unital commutative quantale Q as follows (for clearness sake recall the notion of quantale first). Definition 2. A quantale is a triple (Q,6,⊗) such that • (Q,6) is a join-lattice; • (Q,⊗) is a semigroup; • q ⊗ (∨T) = ∨ t∈T (q ⊗ t) and (∨T) ⊗ q = ∨ t∈T (t ⊗ q) for every q ∈ Q and every T ⊆ Q. A quantale Q is said to be unital provided that there exists an element ∈ Q such that (Q,⊗, ) is a monoid. Q is said to be commutative provided that a ⊗ b = b ⊗ a for every a, b ∈ Q. Definition 3. Let Q be a unital commutative quantale. Q-Alg is the category, the objects of which (called Q-algebras) are unital Q-modules (A, ∗) such that ∗This research was supported by the European Social Fund. 1 • A is a quantale; • q ∗ (a ⊗ b) = (q ∗ a) ⊗ b = a ⊗ (q ∗ b) for every a, b ∈ A and every q ∈ Q. The morphisms of the category (called Q-algebra homomorphisms) are quantale homomorphisms A f− → B which are also Q-module homomorphisms