Some applications of the study of the image of a fuzzy number: Countable fuzzy numbers, operations, regression and a specificity-type ordering

In this paper, we present a characterization of the image of a fuzzy number in terms of the extremes of its level sets. Then we deduce some consequences: we show some relationships between the image of two fuzzy numbers and the images of their usual operations and it is proved that the continuity of some fuzzy numbers is preserved under the usual operations. As applications, we introduce a fuzzy regression model involving a parametric family of fuzzy semidistances between fuzzy numbers with finite image and an ordering process of fuzzy numbers with compact support that satisfies specificity-type properties.