The Schroder-Bernstein property for a-saturated models

The Schröder-Bernstein property for weakly minimal theories

For a countable, weakly minimal theory, we show that the Schroeder-Bernstein property (any two elementarily bi-embeddable models are isomorphic) is equivalent to both a condition on orbits of rank 1 types and the property that the theory has no infinite collection of pairwise bi-embeddable, pairwise nonisomorphic models. We conclude that for countable weakly minimal theories, the Schroeder-Bernstein property is absolute between transitive models of ZFC.

具有 Schröder-Bernstein 性质的一般实值理论是稳定的

General real-valued theories with the Schröder-Bernstein property are stable

本文证明了具有 Schröder-Bernstein 性质的 Keisler 风格一般实值理论是稳定的，推广了经典逻辑中的相应结果，并探讨了该技术在连续理论中的适用性。

The Cantor-Bernstein-Schroder theorem in algebra

The famous Cantor-Bernstein-Schroder 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 establishi…

Borel复杂度与Schröder-Bernstein性质

Borel Complexity and the Schröder-Bernstein Property

本研究提出Borel可归约性的新不变量“厚度”，证明挠阿贝尔群的所有Friedman-Stanley跳跃非Borel完全，并在大基数假设下证明具有Schröder-Bernstein性质的$\u27e8\ucf79_{\u03c9_1 \u03c9}$语句非Borel完全。

The Schroder-Bernstein property for theories of abelian groups

A first-order theory has the Schroder-Bernstein property if any two of its models that are elementarily bi-embeddable are isomorphic. We prove that if G is an abelian group, then the follwing are equivalent: 1. Th(G, +) has the Schroder-Bernstein property; 2. Th(G, +) is omega-stable; 3. G is the direct sum of a divisible group and a torsion group of bounded exponent; 4. Th(G, +) is superstable, and if (H, +) is a saturated elementary extension of (G,+), every map in Aut(H/H^0) is unipotent.

The Cantor-Schröder-Bernstein Theorem for $\infty$-groupoids

We show that the Cantor-Schröder-Bernstein Theorem for homotopy types, or $\infty$-groupoids holds in the following form: For any two types, if each one is embedded into the other, then they are equivalent. The argument is developed in the language of homotopy type theory, or Voevodsky's univalent foundations (HoTT/UF), and requires classical logic. It follows that the theorem holds in any boolean $\infty$-topos.