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Intervals in generalized effect algebras. (English) Zbl 1325.03077
Summary: A significant property of a generalized effect algebra is that its every interval with inherited partial sum is an effect algebra. We show that in some sense the converse is also true. More precisely, we prove that a set with zero element is a generalized effect algebra if and only if all its intervals are effect algebras. We investigate inheritance of some properties from intervals to generalized effect algebras, e.g., the Riesz decomposition property, compatibility of every pair of elements, dense embedding into a complete effect algebra, to be a sub-(generalized) effect algebra, to be lattice ordered and others. The response to the Open Problem from [Z. Riečanová and M. Zajac, “Intervals in generalized effect algebras and their sub-generalized effect algebras”, Acta Polytech. 53, No. 3, 314–316 (2013)] for generalized effect algebras and their sub-generalized effect algebras is given.

MSC:
03G12 Quantum logic
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[1] Blank J, Exner P, Havlíček M (2008) Hilbert space operators in quantum physics, 2nd edn. Springer, Berlin · Zbl 1163.47060
[2] Burns, G, Darstellung und erweiterungen geordneter mengen I, Int J Reine Angew Math, 209, 35-46, (1962)
[3] Chang, CC, Algebraic analysis of many-valued logics, Trans Amer Math Soc, 88, 467-490, (1958) · Zbl 0084.00704
[4] Dvurečenskij A, Pulmannová S (2000) New trends in quantum structures. Kluwer Academic Publishers/Ister Science, Dordrecht/Bratislava · Zbl 0987.81005
[5] Foulis, DJ; Bennett, MK, Effect algebras and unsharp quantum logics, Found Phys, 24, 1331-1352, (1994) · Zbl 1213.06004
[6] Hedlíková J, Pulmannová S (1996) Generalized difference posets and orthoalgebras, Acta Math Univ Comenianae LXV: 247-279 · Zbl 0084.00704
[7] Kalmbach, G; Riečanová, Z, An axiomatization for abelian relative inverses, Demonstratio Math, 27, 769-780, (1996) · Zbl 0826.08002
[8] Kôpka, F; Chovanec, F, D-posets, Math Slovaca, 44, 21-34, (1994) · Zbl 0789.03048
[9] Pullmanová S (1999) Representations of MV-algebras by Hilbert-space effects. Inter J Theor Phys 38:3209-3220 · Zbl 0963.03087
[10] Riečanová Z (2013) Subalgebras, intervals and central elements of generalized effect algebras. Inter J Theor Phys 52:2163-2170 · Zbl 1270.06006
[11] Riečanová Z, Zajac M (2013) Intervals in generalized effect algebras and their sub-generalized effect algebras. Acta Polytechnica · Zbl 1213.06004
[12] Riečanová, Z; Zajac, M; Pulmannová, S, Effect algebras of positive linear operators densely defined on Hilbert spaces, Rep Math Phys, 68, 261-270, (2011) · Zbl 1250.81015
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