CantorBernstein property

The classical Cantor–Bernstein theorem says that two sets X;Y which admit injective mappings X ! Y and Y ! X have the same cardinality. We obtain a di¤erent problem when the sets are equipped with an additional structure which should be preserved by the mappings (isomorphisms). It has been generalized to boolean algebras by Sikorski and Tarski: For any two -complete boolean algebras A and B and elements a 2 A and b 2 B, if B = [0; a]A and A = [0; b]B , then A = B. This result has been generalized to MV-algebras in [8] and [2], to orthomodular lattices in [3] and to more general structures, e.g. e¤ect algebras and pseudo-BCK-algebras, in subsequent papers (see the selected references at the end). There is another line of research, initiated in [3]. Here the question is which algebras satisfy the Cantor–Bernstein theorem in its original form by Sikorski and Tarski, without any additional condition. We summarize some results for orthomodular lattices from [3, 4] and compare them to the situation in MV-algebras.