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Blocks of homogeneous effect algebras. (English) Zbl 0985.03063
An effect algebra is called homogeneous iff $$u\leq v_1\oplus v_2$$, $$u\leq (v_1\oplus v_2)'$$ imply that $$u=u_1\oplus u_2$$ for some $$u_1\leq v_1$$, $$u_2\leq v_2$$. The author proves that every homogeneous effect algebra is the union of all its maximal MV-subalgebras. This generalizes previous results obtained for special classes of homogeneous effect algebras, namely for lattice effect algebras [Z. Riečanová, “Generalization of blocks for $$D$$-lattices and lattice-ordered effect algebras”, Int. J. Theor. Phys. 39, 231-237 (2000; Zbl 0968.81003)] and orthoalgebras [J. Hamhalter, M. Navara and P. Pták, “States on orthoalgebras”, Int. J. Theor. Phys. 34, 1439-1465 (1995; Zbl 0841.03034)]. On the other hand, effect algebras of selfadjoint operators of Hilbert spaces of dimension greater than 1 are not homogeneous.

##### MSC:
 03G12 Quantum logic 06C15 Complemented lattices, orthocomplemented lattices and posets 81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects) 06D35 MV-algebras
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##### References:
 [1] Riečanová, Int. J. Theor. Phys. 39 pp 855– (2000) [2] Giuntini, Found. Phys. 19 pp 769– (1994) [3] DOI: 10.1007/BF02283036 · Zbl 1213.06004 [4] Cohen, An introduction to Hilbert space and quantum logic (1989) · Zbl 0664.46021 [5] Chovanec, Tatra Mt. Math. Publ. 10 pp 1– (1997) [6] Chevalier, Some ideal lattices in partial abelian monoids (1998) [7] DOI: 10.2307/1993423 · Zbl 0093.01104 [8] Bush, Operational quantum physics (1995) [9] Beran, Orthomodular lattices. Algebraic approach (1985) [10] DOI: 10.1007/BF02057883 [11] Riečanová, Tatra Mt. Math. Publ. 16 pp 1– (1998) [12] Pták, Orthomodular structures as quantum logics (1991) [13] Ludwig, Foundations of quantum mechanics (1983) · Zbl 0509.46057 [14] Kôpka, Math. Slovaca 44 pp 21– (1994) [15] Kalmbach, Orthomodular lattices (1983) [16] Jenča, BUSEFAL 80 pp 24– (1999) [17] DOI: 10.1007/s000120050061 · Zbl 0933.03082 [18] DOI: 10.1007/BF01108592 · Zbl 0846.03031 [19] Foulis, Internat. J. Theoret. Phys. 35 pp 789– (1995)
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