Restrictions imposed by the fuzzy extension of relations and functions

Abstract

The fuzzy extension principle has been widely used to extend the domain of mathematical functions and relations from elements of a referential set to fuzzy subsets of that referential set. However, there are restrictions associated with the fuzzy extension principle. This paper addresses the question how restricted is the family of "fuzzy set to fuzzy set mappings" obtained by the fuzzy extension of non-fuzzy functions and relations, as compared to the general family of all possible fuzzy set to fuzzy set mappings. A theorem is presented, with the necessary and sufficient conditions, to determine this restriction for the fuzzy extension of non-fuzzy relations and point-valued functions. It is shown that the fuzzy extension principle would impose a restriction on extended fuzzy set to fuzzy set mappings, which is similar to the linear restriction for point-valued functions. Moreover, the extension of mappings from a set-valued domain to a fuzzy set-valued domain is discussed. It is shown that this extension is well-behaved only for those mappings which preserve subsethood order. Another theorem with the necessary and sufficient condition has been proved to determine the imposed restriction during the fuzzy extension of subsethood order preserving set to set mappings. The two extensions have been compared and several examples are provided.