The Research On The Structure Of A Class Of Generalized Boolean Algebras

With the rapid development of intelligence science and computer theory research,nonclassical logics have become one of the most dynamic research directions in computer science,mathematics,logic and related fields.As is well known,the research of non-classical logics needs to learn from the mature theories and methods of classical propositional logics.As the algebraic systems of classical propositional logics,Boolean algebras have been proved to play an important role in the research of non-classical logics.In order to study a new type of non-classical logic based on quantum computing,quasiBoolean algebras have been introduced as the generalization of Boolean algebras in this paper.By discussing the algebraic structures and topological structures of quasi-Boolean algebras,this kind of logic algebras are studied comprehensively and systematically,which lays a theoretical foundation for further establishing and studying its corresponding logic system.The specific research contents are as follows:First,the definition of a quasi-Boolean algebra is introduced and its basic properties are studied.First of all,the relationships between quasi-Boolean algebras and Boolean algebras,algebras of quasiordered logic,quasi-MV algebras and quasi-BL algebras are discussed.In addition,the special congruences are defined and the properties of its corresponding quotient structures are studied,and then the theorem that any quasi-Boolean algebra can be embedded into the direct product of a Boolean algebra and a flat quasi-Boolean algebra has been proved.Next,the definition of a Boolean quasi-ring is introduced and the relationship between quasiBoolean algebras and Boolean quasi-rings is presented.Finally,injective quasi-Boolean algebras are studied,we prove that every complete Boolean algebra is an injective quasiBoolean algebra.Second,the theory of filters and ideals of a quasi-Boolean algebra is studied.Firstly,the definitions of filters and ideals of quasi-Boolean algebras are given,and then we show that filters and ideals are dual.Moreover,filter congruences are defined and the one-to-one correspondence between the set of filters and the set of filter congruences on a quasi-Boolean algebra is given.Secondly,the properties of filters generated by some subsets,prime filters and maximal filters are investigated.We prove that prime filters and maximal filters are equivalent.Finally,the properties of prime spectra are investigated and the result that the prime spectrum of a quasi-Boolean algebra is a compact Hausdorff topological space is proved.Third,the theory of weak filters and weak ideals of a quasi-Boolean algebra is studied.Firstly,the definitions of weak filters and weak ideals of quasi-Boolean algebras are given and the result that weak filters and weak ideals are dual is proved.In addition,the relationship between weak filters of a quasi-Boolean algebra and filters of a Boolean algebra is discussed.We also present the relationship between filters and weak filters of a quasi-Boolean algebra.Secondly,the properties of weak filters generated by some subsets,prime weak filters,maximal weak filters and weak prime weak filters are studied.Finally,the relationship between filters and weak filters of a quasi-Boolean algebra and filters of the quotient algebra is investigated.Fourth,the theory of reticulation in a quasi-Boolean algebra is studied.The definition of reticulation in a quasi-Boolean algebra is given and the properties of reticulation are investigated.We prove that the prime spectrum of a quasi-Boolean algebra is homomorphic to the prime spectrum of its reticulation.