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The famous Cantor-Bernstein-Schroder theorem (CBS-theorem for short) of settheory was generalized by Sikorski and Tarski to \sigma-complete Booleanalgebras. After this, numerous generalizations of the CBS-theorem, extendingthe Sikorski-Tarski version to different classes of algebras, have beenestablished. Among these classes there are lattice ordered groups, orthomodularlattices, MV-algebras, residuated lattices, etc. This suggests to consider acommon algebraic framework in which various versions of the CBS-theorem can beformulated. In this work we provide this algebraic framework establishingnecessary and sufficient conditions for the validity of the theorem. We alsoshow how this abstract framework includes the versions of the CBS-theoremalready present in the literature as well as new versions of the theoremextended to other classes such as groups, modules, semigroups, rings, *-ringsetc.

The Cantor-Bernstein-Schroder theorem in algebra