Contributions to fuzzy analysis (Q1304270)

The authors present specific notions in order to build an acceptable fuzzy mathematical analysis. Firstly, they define a distance between two fuzzy numbers, here seen as fuzzy subsets of the reals, which becomes a fuzzy metric. After introducing the concept of semivector space and ordered semialgebra, they prove that the set of all fuzzy numbers over the reals is an ordered commutative semialgebra with respect to the scalar multiplication, fuzzy scalar multiplication and fuzzy addition, being the last two operations defined via the Zadeh's extension principle. This semialgebra is extended to an ordered commutative algebra whose elements are called extended fuzzy numbers. So the concept of square root of such ``non-negative'' elements is presented. Successively, a fuzzy norm on a real vector space is given and two suitable examples are studied. Finally, fuzzy topologies generated by fuzzy metrics are introduced using the open fuzzy balls in the sense of the same authors [ibid. 66, No. 2, 137-158 (1994; Zbl 0934.26012)] and of \textit{P. Eklund} and the first author [ibid. 26, No. 3, 333-356 (1988; Zbl 0645.54008)]. It is proved that the above usual operations on fuzzy numbers are indeed fuzzy continuous in such topologies.