Números complexos graduados, ordem local e aplicações

Abstract

Aggregations are functions that have the ability to combine multiple objects into a single object of the same nature. Minimum, maximum, weighted average and arithmetic mean, are examples of aggregations frequently used in everyday life which have several possibilities for applications. However, when working with aggregations, such as those mentioned above, the objects in question are always real numbers. There are almost no studies in the literature that portray these aggregations where objects are complex numbers. This is due to the fact that to introduce some aggregations, the objects involved need to be provided with a total order relation. In this conjecture, one can attest to one of the advantages of obtaining a set with an order relation. The Gradual Complex Numbers (NCG), proposed by the author, was recently applied in the performance evaluation of classification algorithms. The method of evaluating the algorithms based on these numbers was called NCG-method. In proposing this method of evaluation, the need arose to compare the complex gradual numbers used in the process. However, as they did not have an order on these numbers, the author circumvented this need using a calculation where much information about complex gradual numbers is lost. In view of the need and importance of working with sets with a total order relationship, this work introduces a notion on order for complex gradual numbers. In addition, the concept of aggregations on these numbers is presented using the notion of proposed order. Finally, two applications of these new approaches are provided. In the first application, it is shown how the notion of order proposed for complex gradual numbers can be used in order to make the NCGmethod more efficient. The second application consists of showing in a simpler way how aggregations over complex gradual numbers can be used.