Abstract:
Fuzzy implications play an important role in many branches of fuzzy mathematics such as many-valued logics, fuzzy inference and control systems, fuzzy relational equations,image processing and data mining. The so much wide application framework promotes the rapid development of fuzzy implications, among which construction and characterization of new models of fuzzy implications are one of hot research topics. Recently,many scholars proposed several kinds of ordinal sums of fuzzy implications through using the respective Gdel implication or Rescher implication as complements to linear transformations of a given family of fuzzy implications on a given family of subsquares of the unit square. The present paper investigates the sufficient and necessary conditions for a general fuzzy implication to be a proper complement to linear transformations of a given family of fuzzy implications on a given family of subsquares of the unit square, and gives a very general platform for constructing ordinal sums of fuzzy implications which includes as special cases the ordinal sum implications existing in the literature.
