Cantor-Bernstein property for MV-algebras

Abstract

Generalizations of the Cantor-Bernstein theorem have been proved for different types of algebras, starting from σ-complete orthomodular lattices and σ-complete MV-algebras and continuing with more general structures, including (pseudo) effect algebras and (pseudo) BCK-algebras. E.g., for σ-complete MV-algebras a version of the Cantor-Bernstein theorem has been proved which assumes that the bounds of isomorphic intervals are boolean. There is another direction of research which has been paid less attention. We ask which algebras satisfy the Cantor-Bernstein theorem in the same form as for σ-complete boolean algebras, without any additional assumption. In the case of orthomodular lattices, it has been proved that this class is rather large. E.g., every orthomodular lattice can be embedded as a subalgebra or expressed as an epimorphic image of a member of this class. On the other hand, also the complement of this class is large in the same sense. We study the analogous question for MV-algebras and we find out interesting examples of MV-algebras which possess or do not possess this property. This contributes to the investigations of the scope of validity of the Cantor-Bernstein theorem in its original form. © Springer-Verlag Berlin Heidelberg 2007.