A Cantor-Bernstein type theorem for effect algebras. (Q1771957)

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.