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States on systems of sets that are closed under symmetric difference |
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| Authors: | Anna De Simone Mirko Navara Pavel Pták |
| Affiliation: | 1. Department of Mathematics and Applications, University Federico II of Naples, Via Cintia, Naples, Italy;2. +420 22435 7388+420 22435 7385;3. Center for Machine Perception, Department of Cybernetics, Faculty of Electrical Engineering, Czech Technical University in Prague, Prague, Czech Republic;4. +420 224 353 565+420 233 339 238;5. Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University in Prague, Prague, Czech Republic |
| Abstract: | 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 σ‐complete setup and find a large class of such systems with rather interesting state properties. |
| Keywords: | Symmetric difference quantum logic Boolean algebra extension of state Primary: 46L51 Secondary: 28A60 60B99 06C15 06E25 06E75 46N30 81P10 |