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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.

03G12 Quantum logic
Full Text: DOI
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