Let $S_1, S_2$ be structures. Let $S_1 \equiv S_2$ Can we say something interesting about their substructures?
Structures and their substructures.
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first-order-logic
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0To elaborate on Reese's answer, note that we can have a susbtructure $T$ of $S_2$ which is not elementarily equivalent to any substructure of $S_1$. For example, let $S_1$ be the standard model of arithmetic $(\mathbb{N}; +, \times, >)$, $S_2$ be any nonstandard model, $c$ be some infinite element of $S_2$, and $T=\{s\in S_2: S_2\models s>c\}$. Then $T$ is not elementarily equivalent to any substructure of $S_1$ (this is a good exercise). Alternatively, we can take the upper part of any nontrivial cut in $S_2$ to be our $T$, and $T$ will satisfy "$\forall x\exists y(y
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Not much. Certainly every finite substructure of $S_1$ is isomorphic to a finite substructure of $S_2$, and vice versa (because a finite structure can be completely characterized by a formula). Provided both structures are infinite and realize at least one type in common, they also have a common infinite substructure (up to isomorphism) constructed by taking the structure generated by a fixed element of the common type. Beyond that, there isn't really anything else to say, so far as I know.