In this talk we will report our systematic study on k-generated implications induced by continuous multiplicative generators of t-norms over the past five years. The first part will provide a brief overview of their desirable properties including flexible ordering properties, unique determination by natural negations, close connections with nilpotent t-norms, the validity of three classical tautologies. The second part will extend the idea behind k-generated implications to study k-quasi-conjugations of fuzzy implication functions, and study their applications in obtaining new representations and classifications of generator generated implications, and new equational characterizations of the Reichenbach implication. Finally, the third part will outline our very recent work on some of their real-word applications, including suspicious network activity detection,approximate reasoning with the large language model Moodo, fuzzy preference decision making,image enhancement and fuzzy implicative rule mining.