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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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