Modal Operators And Pseudo Valuations On Equality Algebras

The notion of equality algebras was originally introduced by S.Jenei,and it is the kind of logical algebras corresponding to the higher order fuzzy logic.In the study of non-classical logic,modal logic is an important branch.Pseudo valuation theory plays a key role in the study of logical algebraic structures.In this paper,we study the modal operators and pseudo valuations on equality algebras.We will use the results to improve the relevant theories of equality algebra and lay a certain theoretical foundation for the study of other algebraic structures.The main research contents are as follows:First of all,we introduce the concept of modal operator on equality algebras and study some related properties.And we give an equivalent characterization of a mappings on residuated equality algebra being modal operators.Further,we introduce and investigate modal filter and modal congruence on equality al-gebras,and obtain some related results.Moreover,we show that there is one to one correspondence relation between modal filters and modal congruences.Using strong modal filters,we establish the uniform structures on modal equal-ity algebras and we prove that modal equality algebras with uniform topologies are topological modal equality algebras.Secondly,we introduce pseudo valua-tions and(positive)implicative pseudo valuations,and discuss the relationship between them.We provide some conditions for a real-valued function to be a pseudo valuation.And we obtain the relationship between pseudo valuations and filters.By using a pseudo valuation,we induce a congruence relation and further a quotient structure.The specific results are as follows:(1)Let.E be a residuated equality algebra and ?:E?E be a mapping.Then ? is a modal operator on E if and only if,for each x,y ? E,it holds:x?f(y)=f(x)/f(y),f(x?y)?f(x)?f(y).(2)If(E,f)be a modal equality algebra,then there exists one to one correspondence between MF(E,f)and MC(E,f).(3)If(E,f)be a modal equality algebra and ? be an arbitrary family of strong modal filters of E which is closed under intersection,then the structure(E??),T?)is a topological modal equality algebra.(4)We study the pseudo valuations and several different types of pseudo valuations on equality algebras and obtain that a pseudo valuation is a positive implicative pseudo valuation if and only if ?((x ?(x?y))?y)=0;A pseudo valuation is a implicative pseudo valuation if and only if ?(x)=?((x?y)?x).(5)We give the condition for a real-valued function to be a pseudo valuation:? is a pseudo valuation if and only ii x?y?z implies ?(z)<?(x)+?(y)for any x,y,z?E.(6)We prove that the binary relation ?? induced by pseudo valuations is a congruence relation on equality algebras.