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

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.

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