Characterizations of some classes of generated implication solutions to the cross-migrativity

Abstract

This paper focuses on the cross-migrativity between grouping functions and the (g, min)-implications generated by additive generators of continuous Archimedean t-conorms and the $h^{-1}$-implications introduced by generalized additive generators of representable uninorms. In addition, we show that ($g,\min$)-implications and $h^{-1}$-implications do not belong to any of the classes known $(S,N)-,R-,QL-$, generalized-h, h-generated, Yager's f- and g-implications. In particular, we establish a series of necessary and sufficient conditions for the cross-migrativity of the grouping functions and several particular types of generated implications (i.e., h-generated implications, k-generated implications, ($g,\min$)-implications and $h^{-1}$-implications) by using the multiplicative (additive) generators and ordinal sums of grouping functions, the distributive laws and contraction law between grouping functions and fuzzy implications, and study a broader connection between grouping functions and generated implications.