Separable frobenius algebras and completely distributive categories

在数学的许多应用中，有一些感兴趣的对象通过函数或它们的一些泛化而相互关联。为了了解物体，观察和测量是由一些与物体的某些属性相关的实数或复数量组成的。用这样的数字进行计算，用这样的变量进行演绎，得出这样的方程，其数学解扩展了对系统的认识。从感兴趣的对象到与属性选择相关联的数量选择的过程显然是一种抽象。有些信息，人们希望只是不相关的信息，丢失了。

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 +.