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Interval and scale effect algebras. (English) Zbl 0883.03048

A quadruple \((E, \oplus, 0,u)\) consisting of a set \(E\), a partially defined binary operation \(\oplus\), and two elements \(0,u\in E\) is called an effect algebra if \(\oplus\) is commutative and associative, satisfies the zero-one law (i.e. if \(p\oplus u\) is defined, then \(p= 0)\) and the orthosupplement law \((\forall p\in E\), \(\exists!q \in E\) such that \(u= p\oplus q)\). Given a partially ordered abelian group \(G\) and \(u\in G\), the interval \(G^+ [0,u]: =\{p\in G: 0\leq p \leq u\}\) is an effect algebra with \(p\oplus q= p+q\), defined whenever \(p+q\leq u\). Such effect algebras are called interval algebras. In this paper it is shown that for every interval algebra \((E, \oplus, 0,u)\) there exists a partially ordered abelian group \(G\) containing \(E\) such that: \(E=G^+ [0,u]\), every element of \(G\) is a difference of finite sums of elements of \(E\), and every group-valued measure \(\varphi: E\to H\) can be extended to a group homomorphism \(\varphi^*: G\to H\). Such a group \(G\) is called universal ambient group for \(E\). It is also shown that if an effect algebra admits an order-determining set of probability measures, then it is an interval algebra. Finally, scale algebras (i.e. totally ordered effect algebras) are considered. It is shown that a scale algebra \(E\) admits a unique probability measure \(\omega\); the latter is positive (i.e. \(\omega (p)=0\) implies \(p=0)\) iff \(E\) is archimedean. Examples of nonarchimedean scale algebras are also given.

MSC:

03G12 Quantum logic
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
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