The Cantor–Bernstein–Schröder theorem via universal algebra

Freytes, Hector

First

2019-01-01

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Abstract

The Cantor–Bernstein–Schröder theorem (CBS-theorem for short) of set theory was generalized by Sikorski and Tarski to σ-complete Boolean algebras. After this, several 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 the algebraic versions of the CBS-theorem can be formulated. In this work we provide this 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.