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Fuzzy quantum posets and their states. (English) Zbl 0733.60007
A fuzzy quantum poset is a system \(M\subseteq [0,1]^{\Omega}\) such that (i) \(1\in M\); (ii) if \(a\in M\), then \(a^{\perp}:=1-a\in M;\) (iii) 1/2\(\not\in M\); (iv) \(\cup_{i}f_ i\in M\) if \(\min (f_ i,f_ j)\leq 1/2,\) \(i\neq j\), \(f_ i\in M\). A state is a mapping m: \(M\to [0,1]\) such that \(m(a\vee a^{\perp})=1\) for any \(a\in M\); m(\(\cup_{i}a_ i)=\sum_{i}m(a_ i)\) if \(\min (a_ i,a_ j)\leq 1/2.\) The author studies the situation when M possesses at least one state. Some sufficient conditions are presented.

MSC:
60A99 Foundations of probability theory
03E72 Theory of fuzzy sets, etc.
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
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