# Today I Learned

Some of the things I've learned every day since Oct 10, 2016

## 166: Elementary Equivalence and Isomorphism between Finite Structures

Let $\mathcal{L}$ be a first-order language. In general, it is true that if two $\mathcal{L}$-structures $\mathcal{M, N}$ are isomorphic, then they are elementarily equivalent. However, elementary equivalence does not in general imply isomorphism.

When $\mathcal{M, N}$ are finite structures, however, the two conditions are equivalent. That is,

$\mathcal{M} \equiv \mathcal{N} \Leftrightarrow \mathcal{M} \cong \mathcal{N}$.