Definition:Expansion of Structure

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Let $\mathcal L, \mathcal L'$ be signatures of the language of predicate logic.

Let $\mathcal L'$ be a supersignature of $\mathcal L$.

Let $\mathcal A, \mathcal A'$ be structures for $\mathcal L, \mathcal L'$, respectively.

Then $\mathcal A'$ is called an expansion of $\mathcal A$ to $\mathcal L'$ if and only if:

For all function symbols $f$ of $\mathcal L$, one has $f_{\mathcal A'} = f_{\mathcal A}$
For all predicate symbols $p$ of $\mathcal L$, one has $p_{\mathcal A'} = p_{\mathcal A}$

where $f_{\mathcal A'}$ is the interpretation of the function symbol $f$ in the structure $\mathcal A'$.

Also see