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States on systems of sets that are closed under symmetric difference

机译：集在对称差分下闭合的状态

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摘要

We consider extensions of certain states. The states are defined on the systems of sets that are closed under the formation of the symmetric difference (concrete quantum logics). These systems can be viewed as certain set-representable quantum logics enriched with the symmetric difference. We first show how the compactness argument allows us to extend states on Boolean algebras over such systems of sets. We then observe that the extensions are sometimes possible even for non-Boolean situations. On the other hand, a difference-closed system can be constructed such that even two-valued states do not allow for extensions. Finally, we consider these questions in a sigma-complete setup and find a large class of such systems with rather interesting state properties. (C) 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
机译：我们考虑某些状态的扩展。这些状态是在对称差形成（具体的量子逻辑）形成的封闭系统的集合上定义的。这些系统可以看作是某些具有对称差的集合可表示的量子逻辑。我们首先展示紧凑性参数如何使我们能够在此类集合系统上扩展布尔代数上的状态。然后我们观察到，即使对于非布尔值情况，有时也可以进行扩展。另一方面，可以构造差分闭合系统，使得即使是二值状态也不允许扩展。最后，我们在sigma-complete设置中考虑这些问题，并找到了一类具有相当有趣的状态属性的此类系统。 （C）2015 WILEY-VCH Verlag GmbH＆Co.KGaA，Weinheim

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