量子逻辑的内蕴拓扑
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摘要
量子力学和相对论是二十世纪两项最伟大的科学成就.它们的创立和发展不仅导致了一系列重大技术发明,而且使得人们对客观世界的运动规律有了基本正确的革命性的理解.自上世纪九十年代以来,与量子理论相关联的量子计算机、量子信息、量子通讯等理论和技术更是得到了迅猛发展.然而要最终实现有价值的量子计算、量子通讯等,不仅在实用化中存在着巨大困难,而且有的困难甚至是原理性的.从一般原则上讲,这些困难的根源是量子力学的测量问题.经典的冯·诺伊曼测量理论是将每个测量看做一个Hilbert空间上的正交投影算子,从而将研究测量问题转化为研究Hilbert空间上的正交投影算子格.但此格仅能描述可精确测量的量子现象. 1994年, Foulis等人引进了用于描述不可精确测量现象的数学结构,即效应代数,这是量子理论的数学公理化问题的一个重大进展.众所周知,自从扎德创立不确定性数学即模糊数学以来,其思想和方法在计算机、人工智能、控制论等领域得到了重要应用.具有不精确性的效应代数理论有可能将模糊数学与量子理论统一起来.鉴于拓扑理论在计算机科学、逻辑推理、Domain理论中的基础性和核心性作用,本文研究了效应代数的几类典型内蕴拓扑的若干性质及运算连续性问题,主要工作包括如下几方面:
1.在格效应代数中,关于效应代数运算⊕和在区间拓扑下的一元连续性已经得到证明,而⊕和的二元连续性及格运算∧和∨的连续性是否成立仍未知.我们给出例子说明在区间拓扑下效应代数运算⊕的二元连续性不成立,格运算的一元连续性也不成立.本文通过对网在区间拓扑下收敛的刻划,证明了在格效应代数框架下⊕和满足二元连续性的必要条件是其上的区间拓扑是Hausdorff拓扑.同时,给出了效应代数运算和格运算满足二元连续性的充分条件.在第二章的最后,证明了标度效应代数上区间拓扑是Hausdorff拓扑,运算⊕和关于区间拓扑是二元连续的.
2.在效应代数E中,网的序收敛与序拓扑收敛是两个不同概念,当两者一致且E序-连续时,该效应代数称为是序-拓扑的.本文得到,在完备的原子格效应代数E中,下面条件等价: (1) E序-连续, (2) E是序-拓扑的, (3)E是完全序不连通的拓扑格, (4) E是代数的.该结论不但从代数观点,更从拓扑角度说明了原子的格效应代数的性质比一般格效应代数要好得多.此结果将Erne等人的工作从正交模格提升到格效应代数上.同时,本文改进了标度效应代数在序拓扑意义下的一个基本矩阵定理,扩大了该定理的应用范围.关于格效应代数运算在序拓扑下的一元连续性已经得到证明,而二元连续性是否成立还没有结论.本文给出了在完备的序-连续格效应代数上,运算⊕满足二元连续性的必要条件是其上的序拓扑是Hausdorff拓扑,并举例说明了即使在完备的布尔代数上,⊕在序拓扑下也不满足二元连续性.同时,给出了一系列使得运算⊕和满足二元连续性的充分条件.标准算子效应代数是效应代数的典型代表,在量子力学中有重要应用,效应代数一词正是来源于此.标准算子效应代数上的弱算子拓扑和强算子拓扑都是及其重要的拓扑,研究它们和内蕴拓扑的关系是一个有趣的重要内容.本章最后研究了标准算子效应代数上的区间拓扑、序拓扑、弱算子拓扑及强算子拓扑之间的关系,得到如下结果:设WOT和SOT分别是标准算子效应代数E(H)上的相对弱算子拓扑和相对强算子拓扑,τi和τo分别是E(H)上的区间拓扑和序拓扑,则τi≤WOT≤SOT≤τo.
3. Frink理想拓扑是偏序集理论中一类重要的内蕴拓扑.特别地, Frink指出它是链上及有限链乘积上的一类恰当拓扑.但由于其定义的抽象性, Frink理想拓扑在效应代数上的研究远没有区间拓扑和序拓扑一样广泛、深入,甚至关于⊕和的一元连续性也没有得到证明.本文在分配格效应代数上研究了Frink理想拓扑下的运算连续性问题,通过对完全不可约理想和对偶理想的一个直观刻划,用非常巧妙的方法证明了运算∧和∨的二元连续性及、⊕和的一元连续性.本结果蕴含了在布尔代数上,效应代数运算⊕和在Frink理想拓扑下满足二元连续性,而本文分别举例说明了此结论对于区间拓扑和序拓扑都不成立.因此,我们有理由相信,在效应代数理论中, Frink理想拓扑是比区间拓扑和序拓扑更为合适的拓扑.偏序集的Frink理想拓扑之Hausdodrff性是一个有趣而困难的问题,引起了众多学者的广泛兴趣.本文给出了格效应代数上Frink理想拓扑是Hausdodrff拓扑的一个充分条件.关于序拓扑和Frink理想拓扑的关系,本文证明了如下结论:设E是完备的原子分配格效应代数,则Frink理想拓扑强于序拓扑,并且下列条件等价: (1) Frink理想拓扑与序拓扑相同, (2) 1是有限元, (3) E中每个元都是有限元, (4) Frink理想拓扑与序拓扑都是离散拓扑.
Quantum mechanics and principle of relativity are the two greatest achievementsin the twentieth century. Their found and development not only led a series of impor-tant technical inventions but also made people have a almost correct and revolutionarycomprehension about the law of the external world. Since the nineties of last cen-tury the quantum computers, quantum information, quantum communication whichare relative to quantum theory have developed rapidly. While there exists enormousdifficulty in practicability even in theory when people want to achieve the valuablequantum computers, quantum communication, etc. In general, the root of difficultyis measurement. In classical measurement theory of von Neumann, each measure-ment is regarded as a projection of a Hilbert space. Then the study of measurement istransferred to the study of the lattice of projections. However, this kind of lattices justcan describe the sharp phenomena. In 1994, Foulis introduced an algebraic structurefor modeling unsharp measurement which is called an effect algebra. This is a greatdevelopment of the mathematical axiomatization of the quantum theory. As we know,since the uncertainty mathematics, namely fuzzy mathematics was founded by Zadeh,its ideal and methods had important applications in computers, artificial intelligence,cybernetics and so on. The theory of Effect algebras which have the uncertainty maybeunite the fuzzy mathematics and the quantum theory. As topology theory has funda-mental and kernel effect in computer science, logic consequence and Domain theory,we study some properties of several typical intrinsic topologies of effect algebras andoperation continuity with respect to them in this paper. The main work includes thefollowing aspects:
1. The continuity of effect algebraic operations⊕and with respect to one vari-able in the interval topology has been proved. While the continuity of two variablesand the continuity of lattice operations have not been proved. We give an exampleto show that the continuity of effect algebraic operations⊕and of two variablesand the continuity of lattice operations of one variable do not hold with respect tothe interval topology. Following the characteristic of nets convergence in the intervaltopology, we present that the necessary condition to guarantee the effect algebraic op- erations are two variables continuous is that the interval topology is Hausdorff. At thesame time, we give the sufficient conditions such that the operations of effect algebraand lattice are two variables continuous. We also prove that the interval topology ofscale effect algebras is Hausdorff and the effect algebraic operations are two variablescontinuous in the scale effect algebras with respect to the interval topology.
2. In an effect algebra E, the order convergence and the order topology conver-gence are different notions. If these two are coincides and E is (o)-continuous, thenE is called order-topological. We obtain that when E is a complete atomic latticeeffect algebra, the following conditions are equivalent: (1) E is (o)-continuous, (2)E is order-topological, (3) E is a totally order disconnected topological lattice, (4) Eis algebraic. This result shows that atomic lattice effect algebras behave much betterthan arbitrary ones, not only from the algebraic, but also from the topological pointof view. Our result generates Erne’s work from the orthomodular lattices to lattice ef-fect algebras. Also we improve a matrix convergence theorem in scale effect algebrasin the sense of order topology and extend its applicable scope. The effect algebraicoperation continuity of one variable in the order topology has been proved. Whilethe continuity of two variables has not been proved. We prove that in a complete(o)-continuous lattice effect algebra, if⊕is two variables continuous, then the ordertopology is Hausdorff. We give the sufficient conditions to guarantee that the effectalgebraic operations are two variables continuous in the order topology. The standardoperator effect algebra is an typical class of effect algebras and has significant appli-cations in quantum mechanics. The name of effect algebras comes from it. The weakoperator topology and strong operator topology of standard operator effect algebra arevery important. Studying the relationships between them and the intrinsic topologiesis an important and interesting matter. In the last of this chapter, we study the re-lationships between the interval topology、order topology、weak operator topologyand strong operator topology of standard operator effect algebras and obtain the fol-lowing result: Let WOT and SOT be the relative weak operator topology and relativestrong operator topology of standard operator effect algebra E(H) andτi,τo be theinterval topology and order topology of E(H), thenτi≤WOT≤SOT≤τo.
3. The Frink ideal topology is a very important intrinsic topology in the posettheory. Especially, Frink pointed out that it is the correct topology for chains anddirect products of a finite numbers of chains. However, as the definition is abstract, the studies to Frink ideal topology of effect algebras is not so extensive and deepas interval topology and order topology. Even the continuity of one variable of⊕has not been proved. In this paper, we study the continuity in distributive latticeeffect algebras with respect to Frink ideal topology. We prove that∧and∨are twovariables continuous and ,⊕and are one variable continuous. This result impliesthat in Boolean algebras the operations⊕and⊕are two variables continuous in theFrink ideal topology while this conclusion does not hold for interval topology andorder topology. Thus we can believe that Frink ideal topology is more suitable foreffect algebras than interval topology and order topology. In the poset theory, theHausdodrff property of Frink ideal topology is an interesting and difficult problem,and many people are interested in it. We give a sufficient condition such that theFrink ideal topology is Hausdorff. Also, we study the relationship between the ordertopology and the Frink ideal topology and obtain the following result: Let E be acomplete atomic distributive lattice effect algebra. Then the following conditions areequivalent: (1) The order topology and the Frink ideal topology of L are coincide, (2)1 is finite, (3) Each element of E is finite, (4) The order topology and the Frink idealtopology of E are both discrete.
1.在格效应代数中,关于效应代数运算⊕和在区间拓扑下的一元连续性已经得到证明,而⊕和的二元连续性及格运算∧和∨的连续性是否成立仍未知.我们给出例子说明在区间拓扑下效应代数运算⊕的二元连续性不成立,格运算的一元连续性也不成立.本文通过对网在区间拓扑下收敛的刻划,证明了在格效应代数框架下⊕和满足二元连续性的必要条件是其上的区间拓扑是Hausdorff拓扑.同时,给出了效应代数运算和格运算满足二元连续性的充分条件.在第二章的最后,证明了标度效应代数上区间拓扑是Hausdorff拓扑,运算⊕和关于区间拓扑是二元连续的.
2.在效应代数E中,网的序收敛与序拓扑收敛是两个不同概念,当两者一致且E序-连续时,该效应代数称为是序-拓扑的.本文得到,在完备的原子格效应代数E中,下面条件等价: (1) E序-连续, (2) E是序-拓扑的, (3)E是完全序不连通的拓扑格, (4) E是代数的.该结论不但从代数观点,更从拓扑角度说明了原子的格效应代数的性质比一般格效应代数要好得多.此结果将Erne等人的工作从正交模格提升到格效应代数上.同时,本文改进了标度效应代数在序拓扑意义下的一个基本矩阵定理,扩大了该定理的应用范围.关于格效应代数运算在序拓扑下的一元连续性已经得到证明,而二元连续性是否成立还没有结论.本文给出了在完备的序-连续格效应代数上,运算⊕满足二元连续性的必要条件是其上的序拓扑是Hausdorff拓扑,并举例说明了即使在完备的布尔代数上,⊕在序拓扑下也不满足二元连续性.同时,给出了一系列使得运算⊕和满足二元连续性的充分条件.标准算子效应代数是效应代数的典型代表,在量子力学中有重要应用,效应代数一词正是来源于此.标准算子效应代数上的弱算子拓扑和强算子拓扑都是及其重要的拓扑,研究它们和内蕴拓扑的关系是一个有趣的重要内容.本章最后研究了标准算子效应代数上的区间拓扑、序拓扑、弱算子拓扑及强算子拓扑之间的关系,得到如下结果:设WOT和SOT分别是标准算子效应代数E(H)上的相对弱算子拓扑和相对强算子拓扑,τi和τo分别是E(H)上的区间拓扑和序拓扑,则τi≤WOT≤SOT≤τo.
3. Frink理想拓扑是偏序集理论中一类重要的内蕴拓扑.特别地, Frink指出它是链上及有限链乘积上的一类恰当拓扑.但由于其定义的抽象性, Frink理想拓扑在效应代数上的研究远没有区间拓扑和序拓扑一样广泛、深入,甚至关于⊕和的一元连续性也没有得到证明.本文在分配格效应代数上研究了Frink理想拓扑下的运算连续性问题,通过对完全不可约理想和对偶理想的一个直观刻划,用非常巧妙的方法证明了运算∧和∨的二元连续性及、⊕和的一元连续性.本结果蕴含了在布尔代数上,效应代数运算⊕和在Frink理想拓扑下满足二元连续性,而本文分别举例说明了此结论对于区间拓扑和序拓扑都不成立.因此,我们有理由相信,在效应代数理论中, Frink理想拓扑是比区间拓扑和序拓扑更为合适的拓扑.偏序集的Frink理想拓扑之Hausdodrff性是一个有趣而困难的问题,引起了众多学者的广泛兴趣.本文给出了格效应代数上Frink理想拓扑是Hausdodrff拓扑的一个充分条件.关于序拓扑和Frink理想拓扑的关系,本文证明了如下结论:设E是完备的原子分配格效应代数,则Frink理想拓扑强于序拓扑,并且下列条件等价: (1) Frink理想拓扑与序拓扑相同, (2) 1是有限元, (3) E中每个元都是有限元, (4) Frink理想拓扑与序拓扑都是离散拓扑.
Quantum mechanics and principle of relativity are the two greatest achievementsin the twentieth century. Their found and development not only led a series of impor-tant technical inventions but also made people have a almost correct and revolutionarycomprehension about the law of the external world. Since the nineties of last cen-tury the quantum computers, quantum information, quantum communication whichare relative to quantum theory have developed rapidly. While there exists enormousdifficulty in practicability even in theory when people want to achieve the valuablequantum computers, quantum communication, etc. In general, the root of difficultyis measurement. In classical measurement theory of von Neumann, each measure-ment is regarded as a projection of a Hilbert space. Then the study of measurement istransferred to the study of the lattice of projections. However, this kind of lattices justcan describe the sharp phenomena. In 1994, Foulis introduced an algebraic structurefor modeling unsharp measurement which is called an effect algebra. This is a greatdevelopment of the mathematical axiomatization of the quantum theory. As we know,since the uncertainty mathematics, namely fuzzy mathematics was founded by Zadeh,its ideal and methods had important applications in computers, artificial intelligence,cybernetics and so on. The theory of Effect algebras which have the uncertainty maybeunite the fuzzy mathematics and the quantum theory. As topology theory has funda-mental and kernel effect in computer science, logic consequence and Domain theory,we study some properties of several typical intrinsic topologies of effect algebras andoperation continuity with respect to them in this paper. The main work includes thefollowing aspects:
1. The continuity of effect algebraic operations⊕and with respect to one vari-able in the interval topology has been proved. While the continuity of two variablesand the continuity of lattice operations have not been proved. We give an exampleto show that the continuity of effect algebraic operations⊕and of two variablesand the continuity of lattice operations of one variable do not hold with respect tothe interval topology. Following the characteristic of nets convergence in the intervaltopology, we present that the necessary condition to guarantee the effect algebraic op- erations are two variables continuous is that the interval topology is Hausdorff. At thesame time, we give the sufficient conditions such that the operations of effect algebraand lattice are two variables continuous. We also prove that the interval topology ofscale effect algebras is Hausdorff and the effect algebraic operations are two variablescontinuous in the scale effect algebras with respect to the interval topology.
2. In an effect algebra E, the order convergence and the order topology conver-gence are different notions. If these two are coincides and E is (o)-continuous, thenE is called order-topological. We obtain that when E is a complete atomic latticeeffect algebra, the following conditions are equivalent: (1) E is (o)-continuous, (2)E is order-topological, (3) E is a totally order disconnected topological lattice, (4) Eis algebraic. This result shows that atomic lattice effect algebras behave much betterthan arbitrary ones, not only from the algebraic, but also from the topological pointof view. Our result generates Erne’s work from the orthomodular lattices to lattice ef-fect algebras. Also we improve a matrix convergence theorem in scale effect algebrasin the sense of order topology and extend its applicable scope. The effect algebraicoperation continuity of one variable in the order topology has been proved. Whilethe continuity of two variables has not been proved. We prove that in a complete(o)-continuous lattice effect algebra, if⊕is two variables continuous, then the ordertopology is Hausdorff. We give the sufficient conditions to guarantee that the effectalgebraic operations are two variables continuous in the order topology. The standardoperator effect algebra is an typical class of effect algebras and has significant appli-cations in quantum mechanics. The name of effect algebras comes from it. The weakoperator topology and strong operator topology of standard operator effect algebra arevery important. Studying the relationships between them and the intrinsic topologiesis an important and interesting matter. In the last of this chapter, we study the re-lationships between the interval topology、order topology、weak operator topologyand strong operator topology of standard operator effect algebras and obtain the fol-lowing result: Let WOT and SOT be the relative weak operator topology and relativestrong operator topology of standard operator effect algebra E(H) andτi,τo be theinterval topology and order topology of E(H), thenτi≤WOT≤SOT≤τo.
3. The Frink ideal topology is a very important intrinsic topology in the posettheory. Especially, Frink pointed out that it is the correct topology for chains anddirect products of a finite numbers of chains. However, as the definition is abstract, the studies to Frink ideal topology of effect algebras is not so extensive and deepas interval topology and order topology. Even the continuity of one variable of⊕has not been proved. In this paper, we study the continuity in distributive latticeeffect algebras with respect to Frink ideal topology. We prove that∧and∨are twovariables continuous and ,⊕and are one variable continuous. This result impliesthat in Boolean algebras the operations⊕and⊕are two variables continuous in theFrink ideal topology while this conclusion does not hold for interval topology andorder topology. Thus we can believe that Frink ideal topology is more suitable foreffect algebras than interval topology and order topology. In the poset theory, theHausdodrff property of Frink ideal topology is an interesting and difficult problem,and many people are interested in it. We give a sufficient condition such that theFrink ideal topology is Hausdorff. Also, we study the relationship between the ordertopology and the Frink ideal topology and obtain the following result: Let E be acomplete atomic distributive lattice effect algebra. Then the following conditions areequivalent: (1) The order topology and the Frink ideal topology of L are coincide, (2)1 is finite, (3) Each element of E is finite, (4) The order topology and the Frink idealtopology of E are both discrete.
引文
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37 P. A. Marchetti, R. Rubele. Quantum Logic and Non-commutative Geometry.Internat. J. Theoret. Phys. 2007, 46: 51～64
38 Edited by B. Coecke, D. Moore, A. Wilce. Current Research in OperationalQuantum Logic. Algebras, Categories, Languages. Fundamental Theories ofPhysics, 111. Kluwer Academic Publishers, Dordrecht, 2000
39 M. D. Chiara. Reasoning in Quantum Theory: Sharp and Unsharp QuantumLogics(Trends in Logic S). Kluwer Academic, 2004
40 S. Gudder. Open Problems for Sequential Effect Algebras. Internat. J. Theoret.Phys. 2005, 44: 2199～2205
41 S. Gudder. Sequential Products of Quantum Measurements. Rep. Math. Phys.2007, 60: 273～288
42 E. Santos. Quantum Logic, Probability, and Information: The Relation with theBell Inequalities. Internat. J. Theoret. Phys. 2003, 42: 2545～2555
43 D. Mundici. Interpretation of AF C?-Algebras in Lukasiewicz Sentential Calcu-lus. J. Funct. Anal. 1986, 65: 15～63
44 S. Pulmannova. Effect Algebras with the Riesz Decomposition Property and AFC?-Algebras. Found. Phys. 1999, 29: 1389～1401
45 S. Pulmannova. States on Effect Algebras with the Riesz Decomposition Prop-erty and AF C?-Algebras. Proceedings of Europe-SIC’99, Debaets B., Fodor J.and Koczy L.T. eds. 1999, 85～88
46 S. Gudder. Noncommutative Probability and Applications. Real and StochasticAnalysis, 199～238, Trends Math., Birkhauser Boston, Boston, MA, 2004
47 S. Gudder. An Approach to Quantum Probability. Foundations of Probabilityand Physics, 147～160, Quantum Probab. White Noise Anal., 13, World Sci. Pub-lishing, River Edge, NJ, 2001
48 S. Gudder. What Is Fuzzy Probability Theory. Found. Phys. 2000, 30:1663～1678
49 S. Gudder. Structures in Nonstandard Quantum Mechanics. J. Math. Phys. 1994,35: 6332～6343
50 S. Gudder. Foundations of Quantum Probability. Internat. J. Theoret. Phys. 1993,32: 1747～1762
51 J. Pykacz. Fuzzy Quantum Logics and Infinite-Valued Lukasiewicz Logic. In-ternat. J. Theoret. Phys. 1994, 33: 1403～1416
52 P. Bush, M. Grabowski, P. J. Lahiti. Operational Quantum Physics. Springer-Verlag, Berlin, 1995
53 E. G. Beltrametti, S. Bugajski. Effect Algebras and Statistical Physical Theories.J. Math. Phys. 1997, 38: 3020～3030
54 P. Mittelstaedt. Quantum Physics and Classical Physics—In the Light of Quan-tum Logic. Internat. J. Theoret. Phys. 2005, 44: 771～781
55 M. Pavivicic. Quantum Logic for Quantum Computers. Internat. J. Theoret.Phys. 2000, 39: 813～825
56 S. Gudder. Quantum Computational Logic. Internat. J. Theoret. Phys. 2003, 42:39～47
57 S. Gudder. Quantum Computers. Internat. J. Theoret. Phys. 2000, 39:2151～2177
58 S. Gudder. Quantum Computation. Amer. Math. Monthly. 2003, 110: 181～201
59 D. Williams. A New Information Science. Materials Today. 2007, 10: 56～58
60 M. Schulz, S. Trimper. Persistence of Quantum Information. Physics Letters.2007, 368: 351～355
61 A. Delgado, C. Saavedra, J. C. Retamal. Quantum Information and EntanglementTransfer for Qutrits. Physics Letters A. 2007, 370: 22～27
62 A. Nagy. Fisher Information and Steric Effect. Chemical Physics Letters. 2007,449: 212～215
63 J. P. Crutchfield, K. Wiesner. Intrinsic Quantum Computation. Physics A. 2008,372: 375～380
64 M. S. Ying. A Theory of Computation Based on Quantum Logic (I). TheoreticalComputer Science. 2005, 344: 134～207
65 P. Mittelstaedt. Quantum Logic and Decoherence. Internat. J. Theoret. Phys.2004, 43: 1343～1354
66 R. V. Kadison, J. R. Ringrose. Fundamentals of the Theory of Operator Algebras.Vols.Ⅰ-Ⅳ, New York: Harcourt, 1986
67 M. H. Stone. The Theory of Representations for Boolean Algebras. Trans. Amer.Math. Soc. 1936, 40: 37～111
68 H.A. Priestley. Representation of Distributive Lattices by Means of OrderedStone Spaces. Bull. Lon. Math. Soc. 1970, 2: 186～190
69 O. Frink. Topology in Lattices. Trans. Amer. Math. Soc. 1942, 51: 569～582
70 M. Erne. Separation Axioms for Interval Topologies. Proc. Amer. Math. Soc.1980, 79: 185～190
71 Z. Riecanova. Lattice and Quantum Logics with Separated Intervals, Atomiticity.Internat. J. Theor. Phys. 1998, 37: 191～197
72 Z. Riecanova. Order-Topological Lattice Effect Algebras. Contributions to Gen-eral Algebra 15. 2003, 151～159
73 W. B. Qu, J. D. Wu. Continuity of Effect Algebra Operations in the IntervalTopology. Internat. J. Theoret. Phys. 2004, 43: 2311～2317
74 M. Erne, Z. Riecanova. Order-Topological Complete Orthomodular Lattices.Topology and its Applications. 1995, 37: 215～217
75 Z. Riecanova. On Order Topological Continuity of Effect Algebra Operations.Contributions to General Algebra 12. 2000, 349～354
76 J. D. Wu, S. J. Lu, D. H. Kim. Antosik-Mikusinski Matrix Convergence Theoremin Quantum Logics. Internat. J. Theoret. Phys. 2003, 42: 1905～1911
77 J. D. Wu, T. F. Tang, M. H. Cho. Two Variables Operation Continuity of EffectAlgebras. Internat. J. Theoret. Phys. 2005, 44: 581～586
78 Z. Riecanova,武俊德.由可精确测量元控制的效应代数上的态.中国科学(A). 2007, 37: 1377～1384
79 O. Frink. Ideals in Partially Ordered Sets. Ann. Math. 1936, 37: 823～843
80 A. J. Ward. On Relations between Certain Intrinsic Topologies in Partially Or-dered Sets. Proc. Cambriage Philos. Soc. 1955, 51: 254～261
81 C. R. Atherton. Concerning Intrinsic Topologies on Boolean Algebras and Cer-tain Bicompactly Generated Lattices. Glasgow Mathematical Journal. 1970, 11:156～161
82 P. V. Ramana Murty. Ideal Topology on a Distributive Lattice. Jour. Aust. Math.Soc. 1974, 18: 503～508
83 Z. Riecanova. States, Uniformities and Metrics on Lattice Effect Algebras. Int.J. of Uncertainty, Fuzziness, and Knowledge-Based Systems. 2002, 10: 125～133
84 Z. H. Ma, J. D. Wu, S. J. Lu. Ideal Topology on Effect Algebras. Internat. J.Theoret. Phys. 2004, 43: 2319～2323
85麻志浩.量子逻辑上的若干问题.浙江大学博士学位论文. 2006
86虞志坚.量子逻辑的代数结构与运算连续性.浙江大学博士学位论文.2006
87武俊德,周选昌,赵闵亨.标度效应代数上的理想拓扑型收敛定理.中国科学(E). 2006, 36: 593～597
88 E. S. Northam. The Interval Topology of Lattice. Proc. Amer. Math. Soc. 1953,4: 824～827
89 Z. Riecanova. Topological and Order-Topological Orthomodular Lattices. Bull.Austral. Math. Soc. 1993, 47: 509～518
90 S. Pulmannova, Z. Riecanova. Compact Topological Orthomodular Lattices.Contributions to General Algebra 7. 1999, 80: 24～29
91 M. Erne. Z-Continuous Posets and Their Topological Manifestation. AppliedCategorical Structures. 1999, 7: 31～70
92 A. Avallone, P. Vitolo. Lattice Uniformities on Effect Algebras. Internat. J.Theoret. Phys. 2005, 44: 793～806
93 O. Brunet. An Intrinsic Topology for Orthomodular Lattices. Internat. J. Theoret.Phys. 2007, 46: 2887～2900
94 A. Wilce. Symmetry and Topology in Quantum Logic. Internat. J. Theoret. Phys.2005, 44: 2303～2316
95 V. Lazar, I. Marinova. Lattice Effect Algebras with Total Operation. Journal ofElectrical Engineeing. 2001, 52: 59～62
96 Z. Riecanova. Continuous Lattice Effect Algebras Admitting Order-ContinuousStates. Fuzzy Sets and Systems. 2003, 136: 41～54
97 A. Wilansky. Topology for Analysis. Waltham. Mass., Ginn, 1970
98 T. A. Sarymsakov, S. A. Ajupov, D.Chadzijev, V. I. Cilin. Uporiadocennyje Al-gebry. Taskent, 1983
99 D. C. Kent. On the Order Topology in a Lattice. Illinois J. Math. 1966, 10:90～96
100 M. Erne. Order-Topological Lattices. Glasgow Math. J. 1980, 21: 57～68
101 M. Erne, S. Weck. Order Convergence in Lattices. Rocky Mtn. J. Math. 1980,10: 805～818
102 H. Dobbertin. A Note on Order Convergence in Complete Lattices. Rocky Mtn.J. Math. 1984, 14: 647～654
103 H. Kirchheimova, Z. Riecanova. Notes on Order Convergence and Order Topol-ogy. Appendix B, in: B. Riecan, T. Neubrunn (Eds.), Integral, Measure and Order-ing, Kluwer Academic Publishers, Dordrecht and Ister Science, Bratislava, 1997
104 V. Olejcek. The Order Topology on a Lattice and its MacNeille Completion.Internat. J. Theoret. Phys. 2000, 39: 801～803
105 G. Gierz et al. Compendium of Continuous Lattices. Springer Verlag, 1980
106 Z. Riecanova. Smearings of States Defined on Sharp Elements onto Effect Al-gebras. Internat. J. Theor. Phys. 2002, 41: 1511～1524
107 R. L. Li, C. Swartz. Spaces for Which the Uniform Boundedness PrincipleHolds. Studia. Sci. Math. Hungar. 1992, 27: 379～384
108 S. Gudder. Sharply Dominating Effect Algebras. Tatra Mt. Math. Publ. 1998,15: 23～30
109 Z. Riecanova. Basic Decomposition of Elements and Jauch-Piron Effect Alge-bras. Fuzzy Sets and Systems. 2005, 115: 138～149
110 G. Birkhoff, O. Frink. Representation of Sets. Algebra Universalis. 1980, 11:173～192
2 G. Boole. The Laws of Thought. Cambridge, 1854
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22 F. Kopka, F. Chovance. D-Posets. Math. Slovaca. 1994, 44: 21～34
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26 E. G. Beltrametti, M. J. Maczynski. Effect Algebras and Ring-Like Structures.Discuss. Math. Gen. Algebra Appl. 2003, 23: 63～79
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28 A. Dvurecenskij, H. S. Kim. Connections between BCK-Algebras and DifferencePosets. Studia Logica. 1998, 60: 421～439
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30 S. Gudder. Lattice Properties of Quantum Effects. J. Math. Phys. 1996, 37:2637～2642
31 T. Moreland, S. Gudder. Infima of Hilbert Spaces Effects. Linear Algebras andIts Applications. 1999, 286: 1～17
32 A. Gheondea, S. Gudder. Sequential Product of Quantum Effects. Proc. Am.Math. Soc. 2004, 132: 503～512
33 A. Gheondea, S. Gudder, P. Jonas. On the Infimum of Quantum Effects. J. Math.Phys. 2005, 46: 1～11
34 S. Gudder. An Order for Quantum Observables. Math. Slovaca. 2006, 56:573～589
35 H. K. Du, C. Y. Deng, Q. H. Li. On the Infimum Problem of Hilbert SpaceEffects. Science in China: Series A Mathematics. 2006, 49: 545～556
36 S. Pulmannova. Effect Algebras with Compressions. Rep. Math. Phys. 2006,58: 301～324
37 P. A. Marchetti, R. Rubele. Quantum Logic and Non-commutative Geometry.Internat. J. Theoret. Phys. 2007, 46: 51～64
38 Edited by B. Coecke, D. Moore, A. Wilce. Current Research in OperationalQuantum Logic. Algebras, Categories, Languages. Fundamental Theories ofPhysics, 111. Kluwer Academic Publishers, Dordrecht, 2000
39 M. D. Chiara. Reasoning in Quantum Theory: Sharp and Unsharp QuantumLogics(Trends in Logic S). Kluwer Academic, 2004
40 S. Gudder. Open Problems for Sequential Effect Algebras. Internat. J. Theoret.Phys. 2005, 44: 2199～2205
41 S. Gudder. Sequential Products of Quantum Measurements. Rep. Math. Phys.2007, 60: 273～288
42 E. Santos. Quantum Logic, Probability, and Information: The Relation with theBell Inequalities. Internat. J. Theoret. Phys. 2003, 42: 2545～2555
43 D. Mundici. Interpretation of AF C?-Algebras in Lukasiewicz Sentential Calcu-lus. J. Funct. Anal. 1986, 65: 15～63
44 S. Pulmannova. Effect Algebras with the Riesz Decomposition Property and AFC?-Algebras. Found. Phys. 1999, 29: 1389～1401
45 S. Pulmannova. States on Effect Algebras with the Riesz Decomposition Prop-erty and AF C?-Algebras. Proceedings of Europe-SIC’99, Debaets B., Fodor J.and Koczy L.T. eds. 1999, 85～88
46 S. Gudder. Noncommutative Probability and Applications. Real and StochasticAnalysis, 199～238, Trends Math., Birkhauser Boston, Boston, MA, 2004
47 S. Gudder. An Approach to Quantum Probability. Foundations of Probabilityand Physics, 147～160, Quantum Probab. White Noise Anal., 13, World Sci. Pub-lishing, River Edge, NJ, 2001
48 S. Gudder. What Is Fuzzy Probability Theory. Found. Phys. 2000, 30:1663～1678
49 S. Gudder. Structures in Nonstandard Quantum Mechanics. J. Math. Phys. 1994,35: 6332～6343
50 S. Gudder. Foundations of Quantum Probability. Internat. J. Theoret. Phys. 1993,32: 1747～1762
51 J. Pykacz. Fuzzy Quantum Logics and Infinite-Valued Lukasiewicz Logic. In-ternat. J. Theoret. Phys. 1994, 33: 1403～1416
52 P. Bush, M. Grabowski, P. J. Lahiti. Operational Quantum Physics. Springer-Verlag, Berlin, 1995
53 E. G. Beltrametti, S. Bugajski. Effect Algebras and Statistical Physical Theories.J. Math. Phys. 1997, 38: 3020～3030
54 P. Mittelstaedt. Quantum Physics and Classical Physics—In the Light of Quan-tum Logic. Internat. J. Theoret. Phys. 2005, 44: 771～781
55 M. Pavivicic. Quantum Logic for Quantum Computers. Internat. J. Theoret.Phys. 2000, 39: 813～825
56 S. Gudder. Quantum Computational Logic. Internat. J. Theoret. Phys. 2003, 42:39～47
57 S. Gudder. Quantum Computers. Internat. J. Theoret. Phys. 2000, 39:2151～2177
58 S. Gudder. Quantum Computation. Amer. Math. Monthly. 2003, 110: 181～201
59 D. Williams. A New Information Science. Materials Today. 2007, 10: 56～58
60 M. Schulz, S. Trimper. Persistence of Quantum Information. Physics Letters.2007, 368: 351～355
61 A. Delgado, C. Saavedra, J. C. Retamal. Quantum Information and EntanglementTransfer for Qutrits. Physics Letters A. 2007, 370: 22～27
62 A. Nagy. Fisher Information and Steric Effect. Chemical Physics Letters. 2007,449: 212～215
63 J. P. Crutchfield, K. Wiesner. Intrinsic Quantum Computation. Physics A. 2008,372: 375～380
64 M. S. Ying. A Theory of Computation Based on Quantum Logic (I). TheoreticalComputer Science. 2005, 344: 134～207
65 P. Mittelstaedt. Quantum Logic and Decoherence. Internat. J. Theoret. Phys.2004, 43: 1343～1354
66 R. V. Kadison, J. R. Ringrose. Fundamentals of the Theory of Operator Algebras.Vols.Ⅰ-Ⅳ, New York: Harcourt, 1986
67 M. H. Stone. The Theory of Representations for Boolean Algebras. Trans. Amer.Math. Soc. 1936, 40: 37～111
68 H.A. Priestley. Representation of Distributive Lattices by Means of OrderedStone Spaces. Bull. Lon. Math. Soc. 1970, 2: 186～190
69 O. Frink. Topology in Lattices. Trans. Amer. Math. Soc. 1942, 51: 569～582
70 M. Erne. Separation Axioms for Interval Topologies. Proc. Amer. Math. Soc.1980, 79: 185～190
71 Z. Riecanova. Lattice and Quantum Logics with Separated Intervals, Atomiticity.Internat. J. Theor. Phys. 1998, 37: 191～197
72 Z. Riecanova. Order-Topological Lattice Effect Algebras. Contributions to Gen-eral Algebra 15. 2003, 151～159
73 W. B. Qu, J. D. Wu. Continuity of Effect Algebra Operations in the IntervalTopology. Internat. J. Theoret. Phys. 2004, 43: 2311～2317
74 M. Erne, Z. Riecanova. Order-Topological Complete Orthomodular Lattices.Topology and its Applications. 1995, 37: 215～217
75 Z. Riecanova. On Order Topological Continuity of Effect Algebra Operations.Contributions to General Algebra 12. 2000, 349～354
76 J. D. Wu, S. J. Lu, D. H. Kim. Antosik-Mikusinski Matrix Convergence Theoremin Quantum Logics. Internat. J. Theoret. Phys. 2003, 42: 1905～1911
77 J. D. Wu, T. F. Tang, M. H. Cho. Two Variables Operation Continuity of EffectAlgebras. Internat. J. Theoret. Phys. 2005, 44: 581～586
78 Z. Riecanova,武俊德.由可精确测量元控制的效应代数上的态.中国科学(A). 2007, 37: 1377～1384
79 O. Frink. Ideals in Partially Ordered Sets. Ann. Math. 1936, 37: 823～843
80 A. J. Ward. On Relations between Certain Intrinsic Topologies in Partially Or-dered Sets. Proc. Cambriage Philos. Soc. 1955, 51: 254～261
81 C. R. Atherton. Concerning Intrinsic Topologies on Boolean Algebras and Cer-tain Bicompactly Generated Lattices. Glasgow Mathematical Journal. 1970, 11:156～161
82 P. V. Ramana Murty. Ideal Topology on a Distributive Lattice. Jour. Aust. Math.Soc. 1974, 18: 503～508
83 Z. Riecanova. States, Uniformities and Metrics on Lattice Effect Algebras. Int.J. of Uncertainty, Fuzziness, and Knowledge-Based Systems. 2002, 10: 125～133
84 Z. H. Ma, J. D. Wu, S. J. Lu. Ideal Topology on Effect Algebras. Internat. J.Theoret. Phys. 2004, 43: 2319～2323
85麻志浩.量子逻辑上的若干问题.浙江大学博士学位论文. 2006
86虞志坚.量子逻辑的代数结构与运算连续性.浙江大学博士学位论文.2006
87武俊德,周选昌,赵闵亨.标度效应代数上的理想拓扑型收敛定理.中国科学(E). 2006, 36: 593～597
88 E. S. Northam. The Interval Topology of Lattice. Proc. Amer. Math. Soc. 1953,4: 824～827
89 Z. Riecanova. Topological and Order-Topological Orthomodular Lattices. Bull.Austral. Math. Soc. 1993, 47: 509～518
90 S. Pulmannova, Z. Riecanova. Compact Topological Orthomodular Lattices.Contributions to General Algebra 7. 1999, 80: 24～29
91 M. Erne. Z-Continuous Posets and Their Topological Manifestation. AppliedCategorical Structures. 1999, 7: 31～70
92 A. Avallone, P. Vitolo. Lattice Uniformities on Effect Algebras. Internat. J.Theoret. Phys. 2005, 44: 793～806
93 O. Brunet. An Intrinsic Topology for Orthomodular Lattices. Internat. J. Theoret.Phys. 2007, 46: 2887～2900
94 A. Wilce. Symmetry and Topology in Quantum Logic. Internat. J. Theoret. Phys.2005, 44: 2303～2316
95 V. Lazar, I. Marinova. Lattice Effect Algebras with Total Operation. Journal ofElectrical Engineeing. 2001, 52: 59～62
96 Z. Riecanova. Continuous Lattice Effect Algebras Admitting Order-ContinuousStates. Fuzzy Sets and Systems. 2003, 136: 41～54
97 A. Wilansky. Topology for Analysis. Waltham. Mass., Ginn, 1970
98 T. A. Sarymsakov, S. A. Ajupov, D.Chadzijev, V. I. Cilin. Uporiadocennyje Al-gebry. Taskent, 1983
99 D. C. Kent. On the Order Topology in a Lattice. Illinois J. Math. 1966, 10:90～96
100 M. Erne. Order-Topological Lattices. Glasgow Math. J. 1980, 21: 57～68
101 M. Erne, S. Weck. Order Convergence in Lattices. Rocky Mtn. J. Math. 1980,10: 805～818
102 H. Dobbertin. A Note on Order Convergence in Complete Lattices. Rocky Mtn.J. Math. 1984, 14: 647～654
103 H. Kirchheimova, Z. Riecanova. Notes on Order Convergence and Order Topol-ogy. Appendix B, in: B. Riecan, T. Neubrunn (Eds.), Integral, Measure and Order-ing, Kluwer Academic Publishers, Dordrecht and Ister Science, Bratislava, 1997
104 V. Olejcek. The Order Topology on a Lattice and its MacNeille Completion.Internat. J. Theoret. Phys. 2000, 39: 801～803
105 G. Gierz et al. Compendium of Continuous Lattices. Springer Verlag, 1980
106 Z. Riecanova. Smearings of States Defined on Sharp Elements onto Effect Al-gebras. Internat. J. Theor. Phys. 2002, 41: 1511～1524
107 R. L. Li, C. Swartz. Spaces for Which the Uniform Boundedness PrincipleHolds. Studia. Sci. Math. Hungar. 1992, 27: 379～384
108 S. Gudder. Sharply Dominating Effect Algebras. Tatra Mt. Math. Publ. 1998,15: 23～30
109 Z. Riecanova. Basic Decomposition of Elements and Jauch-Piron Effect Alge-bras. Fuzzy Sets and Systems. 2005, 115: 138～149
110 G. Birkhoff, O. Frink. Representation of Sets. Algebra Universalis. 1980, 11:173～192