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.


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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