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}.

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