The Research On Topological Structures And Reticulation Theories Of EQ-algebras

It is well known that fuzzy logics have become an important tool for computer to process uncertain information.Type theory is a kind of higher-order fuzzy logic,while fuzzy type theory is a fuzzy version of higher-order fuzzy logic.From a logical point of view,EQ-algebras are the algebraic semantics of fuzzy logic type theory.From the algebraic point of view,the class of EQ-algebras generalizes the class of residuated lattices.Firstly,we give a new class of EQ-algebras,and improve the filter theory proposed by Nov′ak.Secondly,several kinds of topologies are introduced in EQ-algebras.Finally,in order to establish the relationship between EQ-algebras and classical ordered algebras,we study the reticulation theory on EQ-algebras.The main contents of the study are summarized as follows:In the second chapter,the theory of filters on EQ-algebras is studied.First of all,we introduced new filters on EQ-algebras to improve the filters proposed by Nov′ak in2009.In order to further discuss the properties of filters in EQ-algebras,we introduce a new class of EQ-algebras,namely multiplicatively relative EQ-algebras.Filters have some good properties in multiplicatively relative EQ-algebras.Especially,specific generated formula of filters in multiplicatively relative EQ-algebras is given.We also study local EQ-algebras and give some characterizations of local EQ-algebras.Then,we study the finite direct product of bounded lattice-ordered EQ-algebras,and give the counting formula of cardinal number of maximal and lattice-primed filters in EQalgebras that can be decomposed into finite direct product of the bounded latticeordered EQ-algebras.At the end of this chapter,the two kinds of special filters in EQ-algebras are studied,namely co-annihilators and o-filters.In the third chapter,the topologies of EQ-algebras and topological EQ-algebras are studied.In this chapter,we use two class of family of filters construct two kinds of topologies(namely topologies generated by system of filters and uniform topologies)in EQ-algebras.We discuss the relationship between the topological properties and the algebraic properties on topological EQ-algebras.We prove that both of these topologies can make all of the operations on the EQ-algebras to be continuous,that is,under these topologies,EQ-algebras become topological EQ-algebras.Then,we introduce the concept of convergence of nets in topological EQ-algebras,and study some properties of convergence of nets.In order to discuss the completeness of nets in topological EQ-algebras,we study the profinite completion of EQ-algebras.In the fourth chapter,we study the theory of reticulation in bounded latticeordered EQ-algebras.Firstly,we study topological spaces(prime and maximal spectra spaces)constituted by lattice-primed filters and maximal filters in bounded latticeordered EQ-algebras.Secondly,two kinds of equivalent definitions of reticulation are given in EQ-algebras and some properties of reticulation are studied.Subsequently,we construct reticulation on EQ-algebras in two different ways,and prove that the two reticulations are isomorphic on the same EQ-algebra,that is,in the isomorphic sense,the reticulation is unique.Finally,we study some properties of the reticula functor from category of EQ-algebras to category bounded distributive lattices.