Complemental Fuzzy Sets: A Semantic Justification of $q$-Rung Orthopair Fuzzy Sets

This article introduces complemental fuzzy sets, explains their semantics, and presents a subclass of this model that generalizes intuitionistic fuzzy sets in a novel manner. It also provides practical results that will facilitate their implementation in real situations. At the theoretical level, we define a family of
$\mathfrak {c}$
-complemental fuzzy sets from each fuzzy negation
$\mathfrak {c}$
. We argue that this construction provides semantic justification for all subfamilies of complemental fuzzy sets, which include
$q$
-rung orthopair fuzzy sets (when
$\mathfrak {c}$
is a Yager's fuzzy complement) and the new family of Sugeno intuitionistic fuzzy sets (when
$\mathfrak {c}$
belongs to the class of Sugeno's fuzzy complements). We study fundamental operations and a general methodology for the aggregation of complemental fuzzy sets. Then, we give some specific examples of aggregation operators to illustrate their applicability. On a more practical level, constructive proofs demonstrate that all orthopair fuzzy sets on finite sets that satisfy a mild restriction are Sugeno intuitionistic fuzzy sets, and they are
$q$
-rung orthopair fuzzy sets for some
$q$
too. These contributions produce a new operational model that semantically justifies, and mathematically contains, “almost all” orthopair fuzzy sets on finite sets.