The Cantor-Bernstein-Schroder theorem in algebra

TL;DR

This paper develops a unified algebraic framework for the Cantor-Bernstein-Schroder theorem, extending its applicability across various algebraic structures including groups, rings, and modules.

Contribution

It establishes necessary and sufficient conditions for the theorem's validity in a broad algebraic context, unifying and extending previous results.

Findings

Framework includes existing CBS-theorem versions in literature

Introduces new CBS-theorem versions for additional algebraic classes

Provides criteria for the theorem's applicability across diverse structures

Abstract

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.