Separable frobenius algebras and completely distributive categories

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In many applications of mathematics, there are objects of interest related to each other by functions or some generalization thereof. To understand the objects, observations and measurements are made of some real or complex quantities associated with some attributes of the objects. Calculations with such numbers and deductions with such variables lead to equations whose mathematical solution expands knowledge of the system. The passage from the objects of interest to a selection of quantities associated with a selection of attributes is manifestly an abstraction. Some information, one hopes only irrelevant information, is lost. Sometimes, the calculations that are performed on quantities associated with attributes of objects are possible because calculations are possible with the objects themselves. In such situations nothing is discarded prior to calculation and there is less abstraction. Calculations with objects that are not numbers takes us to a very generalized study of calculation. The rules for calculation, even with operations that look familiar, may be different. Loss of abstraction in favour of increased generality is a characteristic of Category Theory. The first part of this proposal involves a study of the operations and equations found in the classical study of separable Frobenius algebras. The objects to which we will apply such techniques are considerably more encompassing than those that are found in Algebra. However, the conjunction of the separable and Frobenius conditions forces a simplification of the internal structure of an object admitting this external algebraic structure. The second part of this proposal studies a generalization of the equation a x (b + c) = (a x b) + (a x c) to a large class of object operations that generalize x and a similar class of object operations that generalize +.