| 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.
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