论文标题
在晶格有序数值事件的环形结构上
On ring-like structures of lattice-ordered numerical events
论文作者
论文摘要
令S为物理系统的一组状态。当系统在s的不同状态s处定义从s到[0,1]的函数称为数值事件或更准确的s-探行性时，事件的发生的概率p（s）。按部分功能顺序排序的一组S-验证能力产生了所谓的S代数，特别是对晶格排序的代数。其中有从概率理论和希尔伯特空间逻辑中知道的Sigma-Elgebras，它们在量子力学中很重要。任何S频繁的代数都可以用作量子逻辑，并且当这种逻辑被视为布尔代数时，这是特别的兴趣，因为那时观察到的物理系统将是经典的。布尔代数与布尔环（Boolean Rings）一对一的对应关系，并且出现了一个问题，是为了找到一个晶格级代数的模拟对应，以延长布尔代数和布尔语环之间的对应关系。我们通过定义类似环的事件结构（RLSE）来回答这个问题。首先，揭示了RLSE的结构，并在RLSE中表征了布尔斯环。然后，我们确定RLSE如何对应数值事件的晶格定位代数。此外，研究了将S探针的晶格定位代数与RLSE相关联的功能。结果表明，如果将相应的映射限制为基础的矫正晶格上的术语函数，则只有两种方法可以将S-probibilities的晶格订购的代数分配给RLSE。这些术语函数是可以将布尔代数分配给布尔环的函数。
Let S be a set of states of a physical system. The probabilities p(s) of the occurrence of an event when the system is in different states s of S define a function from S to [0,1] called a numerical event or, more accurately, an S-probability. Sets of S-probabilities ordered by the partial order of functions give rise to so called algebras of S-probabilities, in particular to the ones that are lattice-ordered. Among these there are the sigma-algebras known from probability theory and the Hilbert-space logics which are important in quantum-mechanics. Any algebra of S-probabilities can serve as a quantum-logic, and it is of special interest when this logic turns out to be a Boolean algebra because then the observed physical system will be classical. Boolean algebras are in one-to-one correspondence to Boolean rings, and the question arises to find an analogue correspondence for lattice-ordered algebras of S-probabilities generalizing the correspondence between Boolean algebras and Boolean rings. We answer this question by defining ring-like structures of events (RLSEs). First, the structure of RLSEs is revealed and Boolean rings among RLSEs are characterized. Then we establish how RLSEs correspond to lattice-ordered algebras of numerical events. Further, functions for associating lattice-ordered algebras of S-probabilities to RLSEs are studied. It is shown that there are only two ways to assign lattice-ordered algebras of S-probabilities to RLSEs if one restricts the corresponding mappings to term functions over the underlying orthomodular lattice. These term functions are the very functions by which also Boolean algebras can be assigned to Boolean rings.
