Characterizations of Fuzzy Implications Generated by Continuous Multiplicative Generators of T-Norms
Hongjun Zhou · IEEE Transactions on Fuzzy Systems · 2020
In this article, we define the$k$-generated implications by continuous multiplicative generators$k$of$t$-norms, along with the research lines of Yager's$f$-generated implications by continuous additive generators$f$of$t$-norms and$g$-generated implications by continuous additive generators$g$of$t$-conorms and of Balasubramaniam's$h$-generated implications by continuous multiplicative generators$h$of$t$-conorms. This article is mainly motivated by many desirable properties of such generator generating implications for their flexible ordering property, unique determinations by induced natural negations, close relations to nilpotent$t$-norms,$T$-conditionality, the law of importation with more choices of$t$-norms$T$other than$T_{P}$, and good distributivity over$t$-norms and$t$-conorms. The relationships of$k$-generated implications to other well-known classes of fuzzy implications are clarified: they will unify$g$-generated implications with$g(1)0$and hence, have intersections with$(S,N)$-implications; the only intersection with$R$-implications is the Goguen implication and they have no intersections with$f$-generated implications. The main results of this article are several characterizing theorems for (sometimes subclasses of)$k$-generated implications by means of aforementioned desirable properties from the respective perspectives, of which some partially solved or enriched related open problems in the literature. Finally, the superiorities of$k$-generated implications for modeling strict fuzzy preference relations and for constructing novel fuzzy reasoning methods are analyzed.