Generalized Laws of Importation and Cross-Migrativity Based on $(D,N)$-Implications

Abstract:

      In the present article we study the generalized laws of importation and cross-migrativity based on $(D,N)$-implications. On the one hand, we consider the generalized law of importation $I(T(x,\alpha),y)=I(x,J(\alpha,y))$, when $T$ is concretized as a fuzzy conjunction $C$, $I$ and $J$ are concretized as $(D,N)$-implications generated by a fuzzy disjunction $D$ and a fuzzy negation $N$. On the other hand, we discuss the cross-migrativity of fuzzy disjunctions over fuzzy implications, that is, $I(x,D(y,z))=D(y,I(x,z))$, where $I$ is a fuzzy implication, $D$ is a fuzzy disjunction. In addition, we study the relationship between the ($\alpha$-)cross-migrative of $(I,D)$ and the ($\alpha$-)cross-law of importation $(I,I_{D,N},C)$.