Banca de DEFESA: NICOLAS EDUARDO ZUMELZU CARCAMO

Fundamentals of a Fuzzy Mathematical Analysis Based on Fuzzy Numbers and Admissible Orders

Fuzzy numbers, orders on fuzzy numbers, admissible orders, weighted vector-fuzzy graphs, fuzzy Riemann integral, hyperstructures, increasing mean-type functions.

The notion of admissible orders in interval fuzzy logic emerged in 2010 with the aim of providing a minimum criterion that a total order in the set of closed subintervals of the unitary interval [0,1] should meet to be used in applications of this fuzzy theory. Later, this same idea was adapted to other extensions of fuzzy logic. In this thesis, we take the idea of admissible orders outside the context of extensions of fuzzy logic. In fact, here we introduce the notion of admissible order for fuzzy numbers equipped with a partial order, that is, a total order that refines this partial order. We pay special attention to the partial order proposed by Ramík and Rímánek in 1985. Furthermore, we present a method to construct admissible orders over fuzzy numbers from admissible orders defined for intervals, considering a superiorly dense sequence, and we prove that this order is admissible for the order of Ramík and Rímánek. From these admissible orders we study fundamental concepts of Mathematical Analysis in the context of fuzzy numbers. The objective is to take the first steps towards the development of a mathematical analysis of fuzzy numbers in certain admissible orders in a robust and well-founded way, preserving as much as possible properties of traditional mathematical analysis. In this way, we introduce the notion of Riemann integral over fuzzy numbers, called fuzzy Riemann integral, considering admissible orders, and we study properties and characterizations of this integral. We formalize the concepts of vector space without inverses and ordered vector space without inverses, a type of hyperstructures, which generalizes the conventional notion of ordered vector spaces. It is worth noting that the space of triangular fuzzy numbers (TFN) and TFNs with some orders are examples of both hyperstructures. Furthermore, we introduce the notion of increasing functions of average type over fuzzy numbers equipped with admissible orders in general, characterizing them as idempotent, and in particular, in the ordered vector space without inverses. Finally, we introduce the concept of weighted vector-fuzzy graphs and use tools built from average-like functions in the ordered vector space without inverses, to solve types of shortest path problems in weighted graphs.