Several Deductive Systems On Weak MTL-algebras And Their Related Properties

The study on logic algebra systems is an important research direction of the nonclassical mathematical logic, in order to further reveal the characteristics of weak MTL algebras, this research from several aspects to study it again, such as deduc-tive systems, reverse deduction systems, annihilator operators and so on. Firstly, some properties of weak MTL algebras are discussed, and the equivalent definition of weak MTL algebras is obtained, and a kind of new weak MTL algebras-regular weak MTL algebras are introduced. Secondly, the concept of implicative deductive systems and positive implicative deductive systems are introduced into weak MTL algebras, some fundamental properties and the relationships among deductive sys-tems, implicative deductive systems and positive implicative deductive systems are investigated, and some equivalent conditions of implicative deductive systems and positive implicative deductive systems are given. Moreover, the connection between deductive systems and congruence relation is discussed. Finally, the concept of single-point annihilator operator, multiple-point annihilator operator, annihilator, reverse deduction systems and reverse implicative deduction systems are introduced into weak MTL algebras, and the basic properties and the relationship of them have been investigated respectively.The construction of chapters and the concrete contents of this paper are as follows:Chapter 1:Preliminaries. We give the basic concepts and correlation properties of lattice, residual lattice, weak MTL algebras, regular weak MTL algebras. Several characterizations of weak MTL algebras are discussed, and the equivalent definition of weak MTL algebras is revealed.Chapter 2:Several deductive systems and their quotient algebras in weak MTL algebras. At first, the concept of implicative deductive systems and positive implica-tive deductive systems are introduced into weak MTL algebras, and the relationships among deductive systems, implicative deductive systems and positive implicative deductive systems are investigated, and some equivalent conditions of implicative deductive systems and positive implicative deductive systems are given. Finally, the connection between deductive systems and congruence relation is discussed.Chapter 3:Annihilator operators of regular weak MTL algebras. In the first place, the concept of annihilator operator is introduced into regular weak MTL algebras. Then, the properties of single-point annihilator operator and multiple-point annihilator operator have been investigated respectively.Chapter 4:Reverse deduction systems and Annihilator of regular weak MTL-Algebras with their relationships. Firstly, the concept of reverse deduction systems and reverse implicative deduction systems are introduced into regular weak MTL algebras. Secondly, their basic properties are discussed. Finally, the dependent relationship and decision relationship between and reverse deductive systems are explored in this chapter.