# zbMATH — the first resource for mathematics

Basic decomposition of elements and Jauch-Piron effect algebras. (English) Zbl 1073.81014
Summary: We show that every element of a complete atomic effect algebra $$E$$ has a unique basic decomposition into a sum of a sharp element and unsharp multiples of isotropic atoms of $$E$$. Consequently, for such effect algebras we obtain “the Smearing Theorem for states” establishing that every order-continuous state existing on sharp elements of $$E$$ can be extended to a state on $$E$$. For a $$\sigma$$-complete separable atomic effect algebra $$E$$ we prove that E is a unital and Jauch-Piron effect algebra if and only if the set $$S(E)$$ of all sharp elements of $$E$$ is a unital Jauch-Piron orthomodular lattice and for finite $$E$$, $$S(E)$$ is a Boolean algebra.

##### MSC:
 81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects) 03G12 Quantum logic 03B52 Fuzzy logic; logic of vagueness 03E72 Theory of fuzzy sets, etc. 06C15 Complemented lattices, orthocomplemented lattices and posets
Full Text:
##### References:
 [1] Bruce, L.; Navara, M.; Pták, P.; Wright, J., Quantum logics logics with jauch – piron states, Quart. J. math. (Oxford), 36, 261-271, (1985) · Zbl 0585.03038 [2] Dvurečenskij, A.; Pulmannová, S., New trends in quantum structures, (2000), Kluwer Academic Publishers Dordrecht, Boston, London, Ister Science, Bratislava · Zbl 0987.81005 [3] Foulis, D.; Bennett, M.K., Effect algebras and unsharp quantum logics, Found. phys., 24, 1325-1346, (1994) · Zbl 1213.06004 [4] Greechie, R.J., Orthomodular lattices admitting no states, J. combin. theory A, 10, 119-132, (1971) · Zbl 0219.06007 [5] Gudder, S., Sharply dominating effect algebras, Tatra mt. math. publ., 15, 23-30, (1998) · Zbl 0939.03073 [6] Hájek, P., Mathematics of fuzzy logics, (1995), Academic Publishers Dordrecht · Zbl 0865.68113 [7] Jauch, J.; Piron, C., On the structure quantal proposition systems, Helv. phys. acta, 42, 842-848, (1969) · Zbl 0181.27603 [8] G. Jenča, I. Marinová, Z. Riečanová, Central elements, blocks and sharp elements of lattice effect algebras, Proc. 3rd Seminar Fuzzy Sets and Quantum Structures, May 2002, Vyhne, Slovakia, pp. 28-33. [9] Jenča, G.; Riečanová, Z., On sharp elements in lattice ordered effect algebras, Busefal, 80, 24-99, (1999) [10] Kalmbach, G., Orthomodular lattices, (1983), Academic Press London · Zbl 0512.06011 [11] Kôpka, F., D-posets of fuzzy sets, Tatra mt. math. publ., 1, 83-87, (1992) · Zbl 0797.04011 [12] Kôpka, F.; Chovanec, F., Boolean D-posets, Tatra mt. math. publ., 10, 183-197, (1997) · Zbl 0915.03052 [13] Pták, P.; Pulmanová, S., Orthomodular structures as quantum logics, (1991), Veda and Kluwer Academic Publishers Bratislava, Dordrecht, Boston, London [14] Riečanová, Z., On order continuity of quantum structures and their homomorphisms, Demonstratio math., 29, 433-443, (1996) · Zbl 0909.03049 [15] Riečanová, Z., Generalization of blocks for D-lattices and lattice ordered effect algebras, Internat. J. theoret. phys., 39, 231-237, (2000) · Zbl 0968.81003 [16] Riečanová, Z., Archimedean and block-finite lattice effect algebras admitting no states, Demonstratio math., 33, 443-452, (2000) · Zbl 0967.06006 [17] Riečanová, Z., Lattice effect algebras with (o)-continuous faithful valuations, Fuzzy sets and systems, 124, 321-327, (2001) · Zbl 0997.03051 [18] Riečanová, Z., Orthogonal sets in effect algebras, Demonstratio math., 34, 525-532, (2001) · Zbl 0989.03071 [19] Riečanová, Z., Proper effect algebras admitting no states, Internat. J. theoret. phys., 40, 1683-1691, (2001) · Zbl 0989.81003 [20] Riečanová, Z., Smearings of states defined on sharp elements of onto effect algebras, Internat. J. theoret. phys., 41, 1511-1524, (2002) · Zbl 1016.81005 [21] Riečanová, Z., Continuous lattice effect algebras admitting order continuous states, Fuzzy sets and systems, 136, 41-54, (2003) · Zbl 1022.03047
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.