Studies On Effect Algebras And Pseudoeffect Algebras In Quantum Logics

It is well-known that quantum mechanics is a set of rules to construct physical theories, and quantum logics is its mathematical foundation. Since in 1936 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. With the development of the theory of quantum logics, new algebraic structures have been proposed as their models. For example, effect algebras being a quantum model can embody the sharp and unsharp properties while pseudo MV algebras and pseudo effect algebras can meet the requirement of non community on relevant physical systems. In a word, in need of relevant quantum physical systems, there are two classes of methods for studying quantum logics : the operation approach and the algebraic approach. The former is based on the states in physical systems and studies the convex structure of states; the latter is based on the observables in physical systems and studies the algebraic structure of sets consisting of all the observables.In this thesis we mainly focus on two kinds of algebraic structures, i.e. effect algebras and pseudoeffect algebras. These algebraic structures are two main objects in the field of quantum logics. The results obtained in this thesis conduce to the inner constructions of the above two structures, which is also one of the main interests for many researchers. The main contributions in this thesis are cited as follows:(1) From the viewpoint of fuzzy sets on posets, we construct some quantum structures such as effects algebras and σ-effect algebras et al.(2) We first introduce several kinds of effect algebras including the anti-BZ-effect algebras and the S-anti-BZ-effect algebras, and then prove that the set of anti-BZ sharp elements of S-anti-BZ-effect algebras is an orthomodular lattice.(3) We first give the concepts of the generalized ideals and the generalized filters for orthoalgebras, and then extend these concepts to pseudo effect algebras. At last, we attain some good properties on ideals, filters and supports of the above quantum structures.(4) Along the direction of the partial difference, we first give the concepts of pseudo difference posets and pseudo boolean D-posets. Then we prove that a pseudo difference poset is categorical equivalent to a pseudo effect algebra, and a pseudo boolean D-poset is algebraical equivalent to a pseudo MV algebra.(5) To begin with, we introduce the definition of direct limit of pseudo effect algebras, and prove the existence of the direct limit for the category of pseudo effect algebras. Then, we give the definition of the inverse limit for effect algebras, and prove that it is the inverse limit for the category of effect algebras.This thesis is divided into three parts.The first part includes Chapter 1, and mainly reviews recent development on quantum effect structures and quantum pseudo effect structures, and some necessary concepts and results when reading the thesis.The second part consists of Chapters 2, 3 and 4, and mainly introduces the works on quantum effect structures.In Chapter 2, from the stand point of fuzzy sets on posets, we first construct some quantum structures like effect algebras, σ-effect algebras, complete effect algebras, orthoalgebras, orthomodular posets, σ-orthomodular posets and complete orthomodular posets. Then we introduce the concept of fuzzy effect space and establish a representation of a lattice effect algebra with a strong order determining system of states by means of fuzzy effect space.In Chapter 3, we first introduce the definitions of sharply approximating effect algebras, anti-BZ-effect algebras, central approximating effect algebras, and S-anti-BZ-effect algebras. Then we establish not only the relationship between sharply approximating effect algebras and anti-BZ-effect algebras but also the relationship between central approximating effect algebras and anti-BZ-effect algebras. At last we prove that the set of anti-BZ-sharp elements in S-anti-BZ-effect algebras is an orthomodular lattice.In Chapter 4, we first give the definitions of generalized ideals and generalized filters in orthoalgebras, which reflect the structural symmetry on generalized ideals and generalized filters. Secondly, we prove the equivalent relationship between generalized ideals and local ideals, and obtain a characteristic theorem on orthomodular posets by the generalized ideals, that is, a poset is an orthomodular poset iff its principle ideals are generalized ideals. Finally, we establish the connec-...