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States on systems of sets that are closed under symmetric difference
Author(s) -
Simone Anna,
Navara Mirko,
Pták Pavel
Publication year - 2015
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201500029
Subject(s) - mathematics , compact space , class (philosophy) , argument (complex analysis) , state (computer science) , set (abstract data type) , quantum , pure mathematics , closed set , quantum system , discrete mathematics , algebra over a field , computer science , quantum mechanics , algorithm , biochemistry , chemistry , physics , artificial intelligence , programming language
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.