A global view of diverse construction methods of fuzzy implication functions rooted on F-chains
This work provides a theoretical framework for fuzzy logic researchers, but it is incremental as it builds on prior methods without introducing a new paradigm.
The authors tackled the problem of understanding the structural relationships among diverse fuzzy implication functions by generalizing the F-chain-based construction method, showing that it unifies several existing techniques like contraposition and aggregation.
Fuzzy implication functions are one of the most important operators used in the fuzzy logic framework. While their flexible definition allows for diverse families with distinct properties, this variety needs a deeper theoretical understanding of their structural relationships. In this work, we focus on the study of construction methods, which employ different techniques to generate new fuzzy implication functions from existing ones. Particularly, we generalize the $F$-chain-based construction, recently introduced by Mesiar et al. to extend a method for constructing aggregation functions to the context of fuzzy implication functions. Our generalization employs collections of fuzzy implication functions rather than single ones, and uses two different increasing functions instead of a unique $F$-chain. We analyze property preservation under this construction and establish sufficient conditions. Furthermore, we demonstrate that our generalized $F$-chain-based construction is a unifying framework for several existing methods. In particular, we show that various construction techniques, such as contraposition, aggregation, and generalized vertical/horizontal threshold methods, can be reformulated within our approach. This reveals structural similarities between seemingly distinct construction strategies and provides a cohesive perspective on fuzzy implication construction methods.