The Research On Stabilizers And Two Kinds Of Logical Operators On MTL-algebras

Monoidal triangle norm based logic(MTL for short),is an important kind of fuzzy logic,since it was proved that MTL is indeed the logic of all triangular norms.With the deepening of the research of fuzzy logic based on triangular norm,various kinds of logical algebras as the semantical systems of triangular norm logic systems have been extensively introduced and studied.Among these logical algebras,MTL-algebras are the most significant because the others are all particular cases of MTL-algebras.In this thesis,we mainly focus on studying very true operators,similarity operators and stabilizers on MTL-algebras.We want to characterize some subclasses of MTL-algebras and provide an algebraic method for studying truth values of fuzzy propositions in MTL.Also,we want to provide an algebraic foundation for proving the completeness of the corresponding logic systems.The structure of this thesis is as follows:1.In the second chapter,we introduce some classes of stabilizers and investigate related properties of them in MTL-algebras.Then,we also characterize some special classes of MTL-algebras,for example,IMTL-algebras,integral MTLalgebras,G ¨odel algebras and MV-algebras,in terms of these stabilizers.Moreover,we discuss the relation between stabilizers and annihilators in MTL-algebras and obtain that implicative stabilizers and annihilators are equivalent in MTL-algebras.Finally,we discuss the relation among these stabilizers and prove that the right implicative stabilizer and right multiplicative stabilizer are order isomorphic.The results of this chapter also solve two open problems in [Motamed S.,Torkzadeh L..A new class of BL-algebras.Soft Computing,2017,21: 687-698].2.In the third chapter,we introduce very true operators on MTL-algebras and investigate some related properties of them.Also,the conditions for an MTLalgebra to be an MV-algebra and a G ¨odel algebra are given via this operator.Moreover,very true filters on very true MTL-algebras are studied.In particular,subdirectly irreducible very true MTL-algebras are characterized and an analogous of representation theorem for very true MTL-algebras is proved.Moreover,we focus on algebraic structures of the set VF[L] of all very true filters on very true MTL-algebras and obtain VF[L] forms a complete Heyting algebras.Finally,the corresponding logic very true MTL-logic is constructed and the soundness and completeness of this logic are proved based on very true MTL-algebras.3.In the forth chapter,we introduce similarity operators on MTL-algebras and obtain that similarity MTL-algebras.Also,conditions for a similarity operator to be an equivalence are obtained.Then,we discuss the relation between very true operators and similarity operators on MTL-algebras and obtain that the method of mutual conversion between the two operators.Moreover,we introduce and study similarity filters on simialrity MTL-algebras.By using similarity filters,we give some characterizations of representable similarity MTL-algebras.Finally,we introduce the similarity MTL-logic and prove a completeness theorem and conservative extension property of this logic.