Related Properties Of Pseudo-equal Algebras And Their Filters

The structure of logic algebra is an important topic in the study of logic systems.To reveal the characteristics of pseudo equality algebra,this article re-studies it from many aspects.Firstly,the relations between some types of pseudo equality algebras are discussed in detail.Furthermore,we study and show that there are relations among pseudo equality algebras and some of other logical algebras such as pseudo residuated lattice,pseudo MV-algebra and D-lattices.Finally,the structure of the quotient pseudo equality algebras induced by implicative normal deductive systems,fantastic normal deductive systems,prime normal deductive systems and Boolean normal deductive systems are also provided.The main results of this thesis are as follows.1.We introduce the notion of involutive pseudo equality algebras.Moreover,the paper focuses on the properties of bounded,lattice,involutive,prelinear and commutative pseudo equality algebras.Then the relations between these pseudo equality algebras are discussed in detail.2.The relation between pseudo equality algebras and pseudo residuated lattices are studied by using their properties.And it is proved that involutive pseudo equality algebras and involutive pseudo residuated lattices are equivalent.Therefore,pseudo equality algebras can be constructed from pseudo residuated lattices.Secondly,the relations between pseudo equality algebras and pseudo MV-algebras are discussed by using the relations between pseudo residuated lattices and pseudo MV-algebras.We prove that bounded commutative pseudo equality algebras and pseudo MV-algebras are equivalent.Finally,we show that the bounded commutative equality algebra is a D-lattice,and the sufficient condition for constructing a bounded commutative equality algebra by a D-lattice is obtained.3.Based on the extension theorem of implicative deductive systems,we prove that the quotient pseudo equality algebras induced by the implicative closed normal deductive systems of bounded commutative pseudo equality algebras are Boolean algebras.Similarly,based on the extension theorem of fantastic deductive systems,we show that the quotient pseudo equality algebras induced by the fantastic closed normal deductive systems are commutative pseudo equality algebras.Furthermore,the concepts of prime deductive systems and Boolean deductive systems are introduced in pseudo equality algebras,and it is proved that the prime closed normal deductive systems of prelinear pseudo equality algebras are chains,the Boolean closed normal deductive systems induced by bounded commutative pseudo equality algebras are Boolean algebras.