Characterizations of Fuzzy Implications Generated by Continuous Multiplicative Generators of T-Norms

In this article, we define the <tex-math notation="LaTeX">k</tex-math> -generated implications by continuous multiplicative generators <tex-math notation="LaTeX">k</tex-math> of <tex-math notation="LaTeX">t</tex-math> -norms, along with the research lines of Yager's <tex-math notation="LaTeX">f</tex-math> -generated implications by continuous additive generators <tex-math notation="LaTeX">f</tex-math> of <tex-math notation="LaTeX">t</tex-math> -norms and <tex-math notation="LaTeX">g</tex-math> -generated implications by continuous additive generators <tex-math notation="LaTeX">g</tex-math> of <tex-math notation="LaTeX">t</tex-math> -conorms and of Balasubramaniam's <tex-math notation="LaTeX">h</tex-math> -generated implications by continuous multiplicative generators <tex-math notation="LaTeX">h</tex-math> of <tex-math notation="LaTeX">t</tex-math> -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 <tex-math notation="LaTeX">t</tex-math> -norms, <tex-math notation="LaTeX">T</tex-math> -conditionality, the law of importation with more choices of <tex-math notation="LaTeX">t</tex-math> -norms <tex-math notation="LaTeX">T</tex-math> other than <tex-math notation="LaTeX">TP</tex-math> , and good distributivity over <tex-math notation="LaTeX">t</tex-math> -norms and <tex-math notation="LaTeX">t</tex-math> -conorms. The relationships of <tex-math notation="LaTeX">k</tex-math> -generated implications to other well-known classes of fuzzy implications are clarified: they will unify <tex-math notation="LaTeX">g</tex-math> -generated implications with <tex-math notation="LaTeX">g(1)< ∞</tex-math> and <tex-math notation="LaTeX">h</tex-math> -generated implications with <tex-math notation="LaTeX">$h(1)>0$</tex-math> and hence, have intersections with <tex-math notation="LaTeX">$(S,N)$</tex-math> -implications; the only intersection with <tex-math notation="LaTeX">R</tex-math> -implications is the Goguen implication and they have no intersections with <tex-math notation="LaTeX">f</tex-math> -generated implications. The main results of this article are several characterizing theorems for (sometimes subclasses of) <tex-math notation="LaTeX">k</tex-math> -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 <tex-math notation="LaTeX">k</tex-math> -generated implications for modeling strict fuzzy preference relations and for constructing novel fuzzy reasoning methods are analyzed.