Abstract
Given any first-order theory T, it is natural to ask: M and N are models of T such that there are elementary embeddings from M into N and from N into M, is it always true that M and N are isomorphic? If T has this property, then we say that it has the "Shroeder-Bernstein property," or SB. For example, the theory of algebraically closed fields has SB, but the theory of dense linear orders does not. This question may be algebraically interesting in cases where elementary maps have an "algebraic" interpretaion, such as for theories of abelian groups, where they are the pure maps between elementarily equivalent structures. The SB property seems to be related to the stability dichotomies: a theory of abelian groups in the pure language of groups has the SB if and only if it is superstable; and, in general, totally transcendental, non-multidimensional theories have the SB.