On a generalization of multiplicative Sincov's equation for fuzzy implication functions

Characterizations of families of fuzzy implication functions are a necessary step to fully understand the behaviour of the members of the family, their potential applicability and their relations with other families. These characterizations are based on additional algebraical properties that completely define the family. Recently, the class of power based implications was characterized through, among others, the property I(x,y)⋅I(y,z)=I(x,z) in a concrete sub-domain. This property, called in the literature also as multiplicative Sincov's equation, was uncommon to other families, and it was studied in-depth in a previous article. This equality is generalized in this paper by understanding the internal product as the product t-norm and changing it to a general arbitrary continuous Archimedean t-norm. This additional property is analyzed jointly with the weak ordering property or the ordering property, leading to a characterization of those fuzzy implication functions satisfying both additional properties.