Fuzzy numerical functions as special fuzzy multivalued mappings

The concept of a fuzzy numerical function is introduced as a special fuzzy multivalued mapping associating to each element of the universe of discourse a fuzzy (real) number. The notion of a truncated fuzzy number is introduced and its properties are studied. It is proven that the direct image of a fuzzy singleton under a fuzzy numerical function always is a truncated fuzzy number. A strict ordering in the set of truncated fuzzy numbers is constructed and it is shown that the class of fuzzy numbers, endowed with this ordering, forms a lattice. Addition, subtraction, multiplication, division, maximum and minimum of fuzzy numerical functions are defined and the distributivity of the direct image of a fuzzy singleton with respect to these operations is obtained. Carefully chosen definitions of lower and upper semi-continuity of fuzzy numerical functions are presented. The behaviour of addition, subtraction, maximum and minimum of lower (resp. upper) semi-continuous fuzzy numerical functions is investigated.