Affiliations: Department of Electrical and Computer Engineering,
University of Tehran, Tehran 14395, Iran | Department of Electrical Engineering, University of
Texas at Dallas, Richardson, TX 75083, USA
Note: [] Corresponding author: Dr. N. Kehtarnavaz, Department of
Electrical Engineering, EC33, University of Texas at Dallas, PO Box 830688,
Richardson, TX 75083-0688, USA. Tel.: +1 972 883 6838; Fax: +1 972 883 2710;
E-mail: [email protected]
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.
Keywords: fuzzy extension principle, set to set mapping, fuzzy set to fuzzy set mapping, fuzzy relational equation, subsethood order preserving mapping
