Aggregating Fuzzy QL- and (S,N)-Subimplications: Conjugate and Dual Constructions

Renata Hax Sander Reiser, Íbero C. K. Benítez, Adenauer C. Yamin, Benjamín R. C. Bedregal


Fuzzy (S,N)- and QL-subimplication classes can be obtained by a distributive n-ry aggregation  performed over the families of t-subnorms and t-subconorms along with a fuzzy negation. Since these classes of subimplications are explicitly represented by t-subconorms and t-subnorms verifying the generalized associativity, the corresponding (S,N)- and QL-subimplications,  referred as I(S,N) and I_(S,T,N), are  characterized as distributive n-ary aggregation together with related generalizations as the exchange and neutrality principles. Based on these results, the both subclasses I_(S,n) and I_QL of (S,N)- and QL-subimplications which are obtained by the median aggregation operation performed over the standard negation N_S together with  the families  of t-subnorms and t-subconorms S_P and T_P, respectively. In particular, the subclass T_P extends the product t-norm T_P as well as S_P extends the algebraic sum S_P. As the main results, the family of subimplications I_(S_P,N) and I_(S_P,T_P,N) extends the implication class by preserving the corresponding properties. We also present an extension from (S,N)- and QL-subimplications to (S,N)- and QL-implications and discuss dual and conjugate constructions.


Median aggregation; t-sub(co)norms; Fuzzy (sub)implications; QL-implications; (S,N)-implications.

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