Research On The Filter, State And Topological Properties Of Logic Algebra

Fuzzy logical algebras have been proposed as the semantical systems of fuzzy logics.They are studied as a significant aspect of fuzzy propositional logic systems.Establishment of completeness theorem marks harmonious unification of the semantics and syntax of logic systems,while the filter theories play a very key role in proving completeness of the corresponding semantic model algebras.From a logical point of view,different types of filters correspond to the set of different provable formulas in the corresponding propositional logic systems.Therefore the filters play an important role in studying all kinds of logic algebras and they are the main tools to study logic algebras.With the intent of capturing an average degree of truth of a proposition formula of the Lukasiewicz propositional logic system,the notion of states on MV-algebras was proposed by Mundici in 1995,which can be interpreted as the probability of fuzzy events.Moreover,it is an axiomatic generalization of Kolmogorov axioms of classical probability in the multi-valued logic algebras.Bosbach states were intensively intro-duced and investigated by Georgescu on pseudo BL-algebras in 2004.Subsequently,many scholars were committed to the research of states on various kinds of logic al-gebras.So state theories were developed rapidly in more than ten years and many profound and important conclusions on it were obtained.We associate the filter theories with the ideals of relative negations in fuzzy logical algebras.Several types of special filters based on relative negations are proposed in FI algebras and residuated lattices.Furthermore,based on nucleus the involutive filters and two extended forms of them are introduced in residuated lattices.In addition,the notions of Bosbach states and Riecan states on pseudo semihoops are introduced and some properties of them are discussed.Moreover,generalized Bosbach states and generalized Riecan states are also introduced and studied on bounded pseudo semihoops.We try to set up the theories of generalized states on bounded pseudo semihoops.Meanwhile,we also introduce bounded pseudo semihoops with internal states and investigate some properties of internal states intensively.A topology is constructed by upsets and pseudo complement operations of the FI algebra,the properties of this topology are analyzed.Furthermore,a topology is constructed by upsets and nuclei on residuated lattices.In addition,topological properties of the set of prime state filters are introduced on state residuated lattices.And it is proved that topological space of the prime state filters is a compact T0 space.This paper is divided into 5 chapters.Chapter 1 firstly recollects basic results of the FI algebra and semihoop.Secondly,recalls some basic notions and results on residuated lattices and some related logic algebras.Finally,reviews some basic concepts of the general topology.Firstly,the notion of relative negations in FI algebras is proposed and some prop-erties are discussed in chapter 2.Relative regular filters,extended relative regular filters and weakly relative regular filters are defined by using the relative negations and some characteristic theorems of them are obtained.Furthermore,based on rel-ative negations we introduce a extended form of a filter,which is called the set of relative double complemented elements.Then its algebraic properties are introduced and an application of extended relative regular filters is showed.Secondly,a topology is constructed by upsets and pseudo complement operations of the FI algebra.It is proved that this topological space is connected and compact.Finally,the notion of ideals is introduced on FI algebras,and some characteristic theorems of the ideals are obtained.The relationship between the ideals and the filters are established.We also give some examples for specific explanations.Chapter 3 is organized as follows.Firstly,the notion of relatively divisible filters is proposed on residuated lattices and the notion of relative pseudo boolean filters is proposed on MTL algebras.The quotient algebras induced by relative pseudo boolean filters are boolean algebras.We establish the relationship between prime(maximal)fil-ters of residuated lattices and the ones of the set of relative regular elements.Secondly,based on nuclei the notions of involutive filters,extended involutive filters and Glivenko filters are introduced in residuated lattices and some properties of them are discussed.Then some characteristic theorems of them are obtained.Algebraic properties of an extended form of a filter with nucleus are introduced and then an application of ex-tended relative regular filters with nuclei is showed.Finally,a topology is constructed by upsets and nuclei on residuated lattices.It is proved that a residuated lattice with this topology forms a {?,?,(?)}-types semi-topological residuated lattice.Moreover,if a residuated lattice satisfies the Glivenko property,then it with this topology forms a{?}-type left-topological residuated lattice.In chapter 4,We firstly introduce the notion of relative negations on pseudo semihoops and discuss its properties intensively.We introduce relatively additive and orthogonal operations and discusses their properties on pseudo semihoops.Secondly,Bosbach states are introduced on pseudo semihoops and some equivalent forms on it are given.It is proved that there are Bosbach states on bounded perfect pseudo semihoops.Meanwhile,Riecan states are introduced on pseudo semihoops.We prove that Bosbach states are Riecan states on pseudo semihoops but not vice versa,while the converse holds on pseudo semihoops satisfying the Glivenko property.It is also proved that Riecan states on pseudo semihoops is uniquely decided by the ones of the set of relative regular elements.Finally,generalized Bosbach states and generalized Riecan states are introduced on bounded pseudo semihoops.We prove that generalized Bosbach states are generalized Riecan states on bounded pseudo semihoops,and we also prove that generalized Riecan states on bounded pseudo semihoops are reduced to the ones of the set of relative regular elements.In chapter 5,the notion of the internal states firstly introduces on bounded pseudo semihoops and some properties of the internal states are discussed.We give the equiv-alent conditions of bounded pseudo hoops and bounded idempotent pseudo semihoops by the internal states.Meanwhile,Riecan states on bounded good pseudo semihoops can be reduced to the ones of the set of relative regular elements.It is proved that there is a one-to-one correspondence on ?-compatible states of L and Riecan states of ?(L)on bounded good pseudo semihoops L.In addition,we introduce the notions of the state filters and the maximal state filters on bounded good pseudo semihoops.By the maximal state filters,we introduce the notions of local bounded state pseudo semihoops and simple bounded state pseudo semihoops.Then we give some equivalent descriptions of two classes of bounded state pseudo semihoops.Secondly,a sufficien-t and necessary condition is given,under which the set of the state filters becomes a boolean algebra on a state residuated lattice.A lot of characteristic theorems of the prime state filters are obtained on state residuated lattices.the prime state filter theorems are showed on state residuated lattices.Finally,we discuss the topological properties of the set of prime state filters Spec?(L)on state residuated lattices and prove Spec?(L)is a compact T0 space.Furthermore,we discuss the topological prop-erties of the set of maximal state filters Max?(L)when L is a MTL-algebra.And we prove Max?(L)is a compact T2 space.