Generalized homogeneous, prelattice and MV-effect algebras. (English) Zbl 1249.03122

Summary: We study unbounded versions of effect algebras. We show a necessary and sufficient condition for lattice operations of such a generalized effect algebra \(P\) to be inherited under its embeding as a proper ideal with a special property and closed under the effect sum into an effect algebra. Further, we introduce conditions for generalized homogeneous, prelattice and MV-effect algebras. We prove that every generalized prelattice effect algebra \(P\) is a union of generalized MV-effect algebras and every generalized homogeneous effect algebra is a union of its maximal sub-generalized effect algebras with hereditary Riesz decomposition property (blocks). Properties of sharp elements, the center and center of compatibility of \(P\) are shown. We prove that on every generalized MV-effect algebra there is a bounded orthogonally additive measure.


03G12 Quantum logic
06D35 MV-algebras
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
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