Two General Construction Ways Toward Unified Framework of Ordinal Sums of Fuzzy Implications

The present article proposes two construction ways to study the general forms of ordinal sums of fuzzy implications with the intent of unifying the ordinal sums existing in the literature. The first ordinal sum construction way, which we call “Implication Complementing,” is to study how to complement a specific fuzzy implication to the linear transformations of given fuzzy implications defined on respective disjoint subsquares whose principal diagonals are segments of the principal diagonal of the unit square, in order that the resulting ordinal sum on the unit square is a fuzzy implication. The second way, which we call “Implication Reconstructing,” is to study how to reconstruct an initial fuzzy implication through replacing its some values on given rectangular regions of the unit square with the linear transformations of respective given fuzzy implications such that the redefined function is a new fuzzy implication. In both ways, necessary and sufficient conditions for the final reconstructed functions to be fuzzy implications are given and several new constructions of ordinal sums of fuzzy implications are obtained, which would generalize the existing ordinal sums from several aspects. In particular, by adopting the idea behind the second way, the generalized ordinal sums of fuzzy implications are proposed, in which the regions where the linear transformations of given fuzzy implications are defined are neither necessarily subsquares nor necessarily along the principal or minor diagonal of the unit square.