Fuzzy Entails The General Form Of Two Ordered Sums

Fuzzy implications play important roles in many branches of fuzzy mathematics such as many-valued logics,fuzzy inference and control systems,fuzzy relational equations,image processing and data mining.Due to its wide application,fuzzy implications have been widely concerned by many scholars and developed rapidly,especially the construction and characterization of new models of fuzzy implications are one of the hot research topics.Recently,Y.Su,H.W.Liu,P.Drygas,M.Baczyriski et al.proposed several kinds of ordinal sums of fuzzy implications through using the respective Godel implication or Rescher implication as complements to linear transformations of a given family of fuzzy implications.On the one hand,the present thesis 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.On the other hand,the thesis 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 lower triangular parts of subsquares in the unit square,and gives a very general platform for constructing lower triangular ordinal sums of fuzzy implications which includes as special cases all lower triangular ordinal sum implications existing in the literature.The structure of this thesis is organized as follows:1.The general forms of subsquare ordinal sums of fuzzy implications.Firstly,we introduce the notion of closed subsquare ordinal sums of fuzzy implications,and prove that 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 closed subsquares of the unit square.Secondly,we introduce the notion of subsquare ordinal sums of fuzzy implications.At last,we discuss the properties of general subsquare ordinal sums of fuzzy implications and its relation with the existing ordinal sums of fuzzy implications.2.The general forms of lower triangular ordinal sums of fuzzy implications.Firstly,we introduce the notion of closed lower triangular ordinal sums of fuzzy implications,and prove that 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 closed lower triangular parts of subsquares in the unit square.Secondly,we introduce the notion of lower triangular ordinal sums of fuzzy implications.At last,we discuss the properties of lower triangular ordinal sums of fuzzy implications and its relation with the existing ordinal sums of fuzzy implications.