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A Cantor-Bernstein type theorem for effect algebras. (English) Zbl 1061.06020
Effect algebras were introduced by D. J. Foulis and M. K. Bennet in 1994. An effect algebra is a partial groupoid with two constants \(0\) and \(1\) such that the partial binary operation \(\oplus \) is associative, commutative, for each \(a\) there is a unique \(a{'}\) with \(a \oplus a{'} = 1\) and if \(a \oplus 1\) exists then \(a = 0\). It is proved that if \(E_{1}\) and \(E_{2}\) are \(\sigma \)-complete effect algebras such that \(E_{1}\) is a direct factor of \(E_{2}\) and vice versa, then \(E_{1},E_{2}\) are isomorphic.

MSC:
06C15 Complemented lattices, orthocomplemented lattices and posets
03G12 Quantum logic
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
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