The Research Of Some Subclasses And Filter Theory On Pseudo Equality Algebras

Various kinds of logical algebras as nonclassical logical semantic systems have been widely introduced and studied by many scholars.Pseudo equality algebras are the extension of equality algebras,which were originally introduced by S.Jenei and L.Korodi,and renamed JK algebra by A.Dvurecenskij.The filter theory plays a key role in the study of logical algebra structures.In this paper,we will study the special filters and the properties of some subclasses of pseudo equality algebras.The main contents are as follows:Firstly,we give the generating formula of filters on pseudo equality algebras.Secondly,we introduce the notions of(positive)implicative filters and fantastic filters on pseudo equality algebras,study their relations and properties,and dis-cuss the quotient algebras of these special filters.Next,we study some subclasses of pseudo equality algebras,which define the prelinear pseudo equality algebras and divisible pseudo equality algebras.We study the lattice structures of pre-linear pseudo equality algebras,then study the algebraic structure of the set of all filters on divisible pseudo equality algebras.Finally,we research the prop-erties of(?a,?a)-involutive pseudo equality algebras and a-compatible pseudo equality algebras.And we discuss the conditions for involutive pseudo equality algebras to form lattices and commutative pseudo equality algebras.Also,we give the concepts of dense elements and involutive filter of pseudo equality alge-bras.Then by using normal filters,we induce quotient pseudo equality algebras.The specific results are as follows:(1)Let F be a strong normal filter of pseudo equality algebra X.Then F is an implicative filter if and only if F is a fantastic and positive implicative filter.(2)Let F be a normal closed filter of pseudo equality algebra X.Then F is implicative(positive implicative?fantastic)filter if and only if X/F is a implicative(positive implicative?fantastic)pseudo equality algebra.(3)Let X be a prelinear pseudo equality algebra,then X is a distributive lattice.(4)Let X be a divisible pseudo equality algebras,then the set F(X)of all filters can form a Heyting algebra.(5)A pseudo equality algebra X is involutive if and only if(X,?)is a lattice.Moreover,every involutive pseudo equality algebra X is a bounded distributive lattice if for every z ? X,the following hold:x?(y?z)=(x?y)?(x?z)or x?(y?z)=(x?y)=(x?z).(6)Let X be pseudo equality algebra.Then for every a ? X,X is(?a,?a)-involutive if and only if X is commutative and satisfies condition(IV1)or satisfies condition(IV2)as follows,for every x,y?X:(IV1)(x?((y?x)?y))?x=x and(x?(y?(x?y)))?x=x,(IV2)x?(((y?x)?y)?x)=x and x?((y?(x?y))?x)=x.(7)A normal closed filter F of pseudo equality algebra X is involutive if and only if X/F is involutive pseudo equality algebra.(8)If X is a 0-compatible symmetric pseudo equality algebra,then X/Den(X)is a involutive pseudo equality algebra.(Den(X)={x?X|0?x=x?0=0} is the set of all dense elements of X).