We propose a generalization of trapezoidal fuzzy numbers based on modal interval theory, which we name “modal interval trapezoidal fuzzy numbers”. In this generalization, we accept that the alpha cuts associated with a trapezoidal fuzzy number can be modal intervals, also allowing that two interval modalities can be associated with a trapezoidal fuzzy number. In this context, it is difficult to maintain the traditional graphic representation of trapezoidal fuzzy numbers and we must use the interval plane in order to represent our modal interval trapezoidal fuzzy numbers graphically. Using this representation, we can correctly reflect the modality of the alpha cuts. We define some concepts from modal interval analysis and we study some of the related properties and structures, proving, among other things, that the inclusion relation provides a lattice structure on this set. We will also provide a semantic interpretation deduced from the modal interval extensions of real continuous functions and the semantic modal interval theorem. The application of modal intervals in the field of fuzzy numbers also provides a new perspective on and new applications of fuzzy numbers.

In this paper, we bring together two concepts related to uncertainty and vagueness: fuzzy numbers and intervals. With them, we build a new structure whose elements we call interval fuzzy segments. We have undertaken this based on the conviction that the fuzzy numbers are a correct representation of the real numbers under situations of indeterminacy. We also believe that if it makes sense to consider the set of real numbers between two real bounds, then it also makes sense to consider the set of all the fuzzy numbers between two fuzzy number bounds. In this way, we extend the concept of real interval to the concept of interval fuzzy segment defined by two fuzzy bounds and a transition mapping that leads from the lower fuzzy bound to the upper fuzzy bound and this transition mapping generates the set of all the fuzzy numbers comprised between those fuzzy bounds. At the same time, this transition mapping brings the concept of interval fuzzy segment closer to the concept of line segment.

The aim of this work is to construct quantified trapezoidal fuzzy numbers as an extension of trapezoidal fuzzy numbers, by using modal intervals and accepting the possibility that the α-cuts of a trapezoidal fuzzy number may also be improper intervals. In addition, this paper addresses the inclusion relationship which is deduced from the inclusion of modal intervals and is related to the classical set-inclusion relationship between trapezoidal fuzzy numbers. Moreover, in this paper we also study the extensions of real continuous functions over the set of quantified trapezoidal fuzzy numbers. Using the semantic interpretation of the calculations over modal intervals will enable us to interpret the meaning of the calculus accurately over quantified trapezoidal fuzzy numbers. With quantified trapezoidal fuzzy numbers, we will be able to overcome some operational limitations that are usually faced when working with trapezoidal fuzzy numbers from a classical point of view. In order to show the applicability of quantified trapezoidal fuzzy numbers, we propose fuzzy equations which have no solution in the set of proper fuzzy numbers yet do have solutions that are improper fuzzy numbers. We also propose two applications of quantified trapezoidal fuzzy numbers, one of them about financial calculations and the other one in an optical problem.