On coproducts of quantale algebras

While the study of quantale-like structures goes back up to the 1930’s (notwithstanding that the term itself was introduced in [2] 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. In particular, one can enunciate the following slogan: Quantales are to linear logic as frames are to the intuitionistic one. This talk is meant as a contribution to the theory of quantales. We begin by introducing the category Q-Alg of algebras over a given unital commutative quantale Q (shortly Q-algebras). By analogy with monoid rings of [1] we construct a free Q-algebra from a given semigroup. It follows that the category Q-Alg is a monadic construct and therefore is cocomplete. In [3] the authors give a characterization of coproducts in the category Frm of frames through the notion of frame nucleus using a technique of producing quotient frames by means of extending a relation (identifying elements according to the needs of a construction) to a congruence. Following the example we introduce the notion of quantale algebra nucleus which generalizes the notions of quantic nucleus and module nucleus [4]. The concept gives rise to a partial generalization of the aforesaid result for frames to the case of Q-algebras.