Partial T-norms And Related Partial Algebraic Systems

T-norm is a binary operation defined on[0,1],which plays an important role in the study of formal systems of fuzzy logic.In 2018,Borzooei put forward the concept of partial t-norms and extended the binary operations in t-norms to partial binary operations,thus providing a new perspective for studying fuzzy logic.On the one hand,partial t-norms are closely related to quantum effect algebra,and they are also related to fuzzy partial logic.Fuzzy partial logic was first developed by Běhounek et al.proposed and described the "undefined" situation(a special uncertainty)in practical applications.In fuzzy logic,t-norms are closely related to fuzzy implication,but in fuzzy partial logic,there is little research on fuzzy implication induced by partial t-norms.This thesis is mainly analogous to the analytical method of t-norms,studying the fuzzy implication and partial algebraic structure related to partial t-norms,enriching the algebraic system.In this thesis,partial t-norms and partial algebraic structures are studied systematically,the concept of partial fuzzy implication is introduced for the first time,and partial residuated implication induced by partial t-norms is studied.On this basis,three special partial t-norms are studied respectively:regular partial t-norms,left continuous partial t-norms and continuous partial t-norms;This thesis points out the defects of the concept of partial residuated lattice(called ZL-partial residuated lattice in this thesis)in the existing literature,gives a new definition of partial residuated lattice,and discusses the relationship between partial residuated lattice and algebraic systems such as residuated lattice and quasiresiduated lattice;Some special partial residuated lattices,such as regular partial residuated lattices,partial MTL-algebras and partial BL-algebras(they are extensions of two important types of fuzzy logic algebras-MTL-algebras and BL-algebras on partial residuated lattices)are proposed.The properties and filter theory of these algebraic systems are studied.The main results in this thesis are as follows:(1)The definition of partial fuzzy implication and the general form of partial residuated implication induced by partial t-norms are given for the first time;Based on partial t-norms,the concept of residuated lattice is extended,and a reasonable definition of partial residuated lattice is proposed for the first time;The sufficient conditions for a lattice effect algebra to become a partial residuated lattice and a partial residuated lattice to become a residuated lattice are given;The relations between quasiresiduated lattices,partial residuated lattices and regular partial residuated lattices are analyzed;The concept of partial t-conorm is introduced,and it is proved that ZL-partial residuated lattice is co-residuated lattices;Starting from any strong filter of the partial residuated lattice,the quotient structure of the corresponding partial residuated monoid is constructed.(2)The concept of the regular partial t-norm is proposed,and the expression of the regular partial residuated implication induced by the regular partial t-norm is obtained;The definition of regular partial residuated lattice is given and its properties are studied;The transformation conditions between commutative Q-residuated lattices and partial residuated lattices,and the condition for commutative quasiresiduated lattices to become regular partial residuated lattices are obtained.In order to study the quotient structure of regular partial residuated lattices,a special regular partial residuated lattice is introduced,and the Good filter is defined,and the quotient structure of regular partial residuated lattices is constructed.(3)It is proved that the partial algebraic structure corresponding to the continuous partial t-norm and its partial residuated implication is a partial BL-algebra,and its properties are studied;This thesis reveals the filter theory of regular partial BL-algebras and regular partial MTL-algebras,gives the definitions of various filters:(regular)partial BL-filters and(regular)partial MTL-filters,and characterizes(regular)partial BL-algebras and(regular)partial MTL-algebras with corresponding filters;The relationship between some partial algebraic structures involved in this thesis is comprehensively discussed;The concept of Well regular partial BLfilter is given,and the quotient structure of regular partial BL-algebra is constructed successfully.