The Cantor–Bernstein–Schröder theorem via universal algebra

Abstract

The famous Cantor-Bernstein-Schröder theorem (CBS-theorem for short) of set theory was generalized by Sikorski and Tarski to σ-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.

Norms on Categories and Analogs of the Schröder-Bernstein Theorem

“…Though, there are some notable exceptions, which include measure spaces [Sri98,§ 3.3] and a noncommutative version thereof for von Neumann algebras [KR97, Proposition 6.2.4]. Efforts to generalize the CBS theorem in alternative set-ups have recently revived, including results for categories of universal algebras [Fre19] and in homotopy type theory and boolean ∞-topoi [Esc20b] including a formalization in Agda [Esc20a]. Of course, one can also weaken the property by replacing "monomorphism" by some stronger or related notion of morphism.…”