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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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##### References:
  Beltrametti, E.; Cassinelli, G., The logic of quantum mechanics, Encyclopaedia of mathematics and its applications, (1981), Addison-Wesley Reading · Zbl 0504.03026  Birkhoff, G., Lattice ordered groups, Ann. math., 43, 298-331, (1942) · Zbl 0060.05808  Busch, P.; Lahti, P.; Mittelstaedt, P., The quantum theory of measurement, Lecture notes in physics, new series M2, (1991), Springer-Verlag Berlin  Darnel, M.R., Theory of lattice-ordered groups, (1995), Dekker New York · Zbl 0810.06016  Foulis, D.J.; Randall, C.H., What are quantum logics and what ought they to be?, (), 35-52  Foulis, D.J.; Bennett, M.K., Effect algebras and unsharp quantum logics, Found. phys., 24, 1331-1352, (1994) · Zbl 1213.06004  Foulis, D.J.; Greechie, R.J.; Bennett, M.K., Sums and products of interval algebras, Internat. J. theoret. phys., 33, 2119-2136, (1994) · Zbl 0815.06015  Fuchs, L., Partially ordered algebraic systems, International series of monographs on pure and applied mathematics, 28, (1963), Pergamon Oxford  Giuntini, R.; Greuling, H., Toward a formal language for unsharp properties, Found. phys., 19, 931-945, (1989)  K. Goodearl, Partially Ordered Abelian Groups with Interpolation, Mathematical Surveys and Monographs, 20, Am. Math. Soc. Providence, RI · Zbl 0589.06008  Goodearl, K.; Handleman, D., Rank functions andK0, J. pure appl. algebra, 7, 195-216, (1976)  Gudder, S., Quantum probability, (1988), Academic Press Boston · Zbl 0652.60004  Hamhalter, J.; Navara, M.; Pták, P., States on orthoalgebras, Internat. J. theoret. phys., 34, 1439-1465, (1995) · Zbl 0841.03034  Kôpka, F.; Chovanec, F., D, Math. slovaca, 44, 21-34, (1994) · Zbl 0789.03048  Ludwig, G., Foundation of quantum mechanics, (1983/1985), Springer-Verlag New York  Ludwig, G., An axiomatic basis for quantum mechanics, (1986/1987), Springer-Verlag New York  Pták, P.; Pulmannová, S., ()  Piron, C., ()  Prugovecki, E., Stochastic quantum mechanics and quantum space time, (1986), Reidel Dordrecht  Schroeck, F.; Foulis, D., Stochastic quantum mechanics viewed from the language of manuals, Found. phys., 20, 823-858, (1990)  Wilce, A., Partial abelian semigroups, Internat. J. theoret. phys., 34, 1807-1812, (1955) · Zbl 0839.03047
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