T-superiority and t-norm-based images of fuzzy sets

Abstract

Comparisons of finite fuzzy sets  based on the results of conjunctive and disjunctive aggregations of their membership grades are studied.
For a given t-norm $T$, the  notion of $T$-superiority  of fuzzy sets is introduced. When a fuzzy set $g$ is $T$-superior to a fuzzy set $f$, the description of a vague concept by $g$ is much more desirable than the description by $f$.
For a comparison of  a fuzzy set  $f$ with the  ``standard'' fuzzy set $g$  (describing the desirable or historically the most common characterization of the vague concept in question)   t-norm-based transformations of $f$ with respect to $g$, called the $T$-images of $f$, are suggested.  A special type of  the $T$-image of $f$ with respect to $g$ may be considered as a soft evaluation of $T$-superiority of $g$ to $f$.