THE FROBENIUS RELATIONS MEET LINEAR DISTRIBUTIVITY

The notion of Frobenius algebra originally arose in ring theory, but it is a fairly easy observation that this notion can be extended to arbitrary monoidal categories. But, is this really the correct level of generalisation? For example, when studying Frobenius algebras in the *-autonomous category Sup, the standard concept using only the usual tensor product is less interesting than a similar one in which both the usual tensor product and its de Morgan dual (par) are used. Thus we maintain that the notion of linear-distributive category (which has both a tensor and a par, but is nevertheless more general than the notion of monoidal category) provides the correct framework in which to interpret the concept of Frobenius algebra.