  The famous Cantor-Bernstein-Schroder theorem (CBS-theorem for short) of set
theory was generalized by Sikorski and Tarski to \sigma-complete Boolean
algebras. After this, numerous generalizations of the CBS-theorem, extending
the Sikorski-Tarski version to different classes of algebras, have been
established. Among these classes there are lattice ordered groups, orthomodular
lattices, MV-algebras, residuated lattices, etc. This suggests to consider a
common algebraic framework in which various versions of the CBS-theorem can be
formulated. In this work we provide this algebraic framework establishing
necessary and sufficient conditions for the validity of the theorem. We also
show how this abstract framework includes the versions of the CBS-theorem
already present in the literature as well as new versions of the theorem
extended to other classes such as groups, modules, semigroups, rings, *-rings
etc.
