Pseudo-effective Algebraic Pass Fuzzy Topological Space

The paper does some research not only on effect algebras and pseudoeffect algebras but also on L-topological spaces.But all ideas of the research come from lattice-valued fuzzy topology.It is well-known that quantum logics is the mathematical foundation of qunatum mechanics which is a set of rules to construct physical theories.Since 1936 when G. Birkhoff and J.von Neumann proposed the concept of quantum logics,the lattice of all closed subspaces of a separable infinite dimensional complete Hilbert space, as an orthomodular lattice,has been regarded as a main mathematical model for a calculus of quantum logics.However,with the development of the theory of quantum logics,two kinds of algebraic structures,effect algebras and pseudoeffect algebras, become the the main objects which are studied in the field of quantum logics.In the first part of the paper,algebraic properties of effect algebras and pseudoeffect algebras were mainly studied from the angle of lattice theory.Compared with classic topological space,fuzzy topological space contains an extra structure:the layer structure.So the establishment of a theory in fuzzy topology is much more difficult than in the corresponding classical topology theory. On the unit interval issue is the case:from Hutton's first fuzzy unit interval I(L) to liu and Luo's fuzzy unit interval I^*(L),and then to Wang and Xu's H(λ) unit interval,they all have their own shortcomings.Therefore,Wang pointed out in his literature[23]:" How to construct a better standard unit interval is well worth exploring ".Following the clue that " How to construct a better standard unit interval",we do some research in fuzzy topology in the second part of the paper.The results in this paper consist of the following statements:(1) We introduce the concepts of strong congruences,Riesz strong congruences, normal weak Riesz ideals and weakly algebric subsets in the pseudoeffect algebras and discuss their properties in detail.Then we establish the order isomorphism relation between the lattice of all Riesz strong congruences and the lattice of all normal weak Riesz ideals.And we prove that a quotient of a lattice ordered pseudoeffect algebra E with respect to a normal weak Riesz ideal I is linearly ordered if and only if I is a prime normal weak Riesz ideal,and this quotient is a pseudo MV-algebra if and only if I is an intersection of prime normal weak Riesz ideals.Moreover,we give the equivalent characterization of weakly algebric subsets in the pseudoeffect algebras and show that weakly algebric subsets of a pseudoeffect algebra are in a one-to-one correspondence with normal weak Riesz ideals.(2) We dicuss some properties of compressible effect algebra,and answer an open question aeked in S.Gudder's paper[18]:Is the cartesian product of an infinite number of E_i a compressible effect algebra if and only if each E_i is a compressible effect algebra? Moreover.we introduce the concepts of well inside relation,regular elements and normal elements in effect algebras and show that C(E)(?)N(E)(?) P(E),that R(E) and N(E) are all normal sub-effect algebras of E,and that N(E) is an orthomodular poset eta,where C(E),N(E),R(E) and P(E) are respectively the sets of all center elements,all normal elements,all regular elements and all principle elemente of an effect algebra E.(3) We first discuss the basic properties of L-Lowen spaces.Then we introduce the Lowenficition of L-topological spaces,and then discuss the compactification of this kind of space in which the exterior Lowenfication space I^e(L) of Hutton unit interval I(L) is treated as the standard unit interval.Moreover,we discuss the properties of induced I(L) topological spaces of induced space,weakly induced space and L-Lowen space.(4) We first introduce the concept of(IC) spaces and discuss the basic properties of(IC) spaces.Then we introduce the(IC)-ficition of L-topological spaces,and establish the imbedding theorem and(IC) Stone-(?)ech compactification of this kind of space in which the exterior(IC)-fication space(?)(L) of Hutton unit interval I(L) is treated as the standard unit interval.Moreover,we introduce the concept of (IC)LM-fuzzy topological spaces and discuss its basic properties and categorical properties and obtain some nice results.(5) We first introduce the concept of k-L-spaces and k_R-L-spaces,and prove that [0,1]-topological spaces(X,δ) is k-[0,1]-spaces if and only if(X,δ) is the quotient of local ultra-F₁ compactness spaces and the product space of k-[0,1]-space(X,δ) and local ultra-F₁ compactness space(Y,η) is k-[0,1]-space etc.Moreover,by introducing ultra-F₁ compact-open topology,we prove UF₁T₂k-[0,1]-Top(the category of all ultra-F₁ T₂ k-[0,1]topological spaces and L-continuous mapping) and k_R-[0,1]-ST₀CReg^-(the category of all sub-T₀(IC) complete regularity k_R-[0,1]topological spaces and L-continuous mapping) are all Cartesian closed category.