# Definition:Reduct of Structure

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## Definition

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

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

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

Then $\mathcal A$ is called the reduct 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 $\mathcal A'$.

Symbolically, one may write $\mathcal A = \mathcal A' \restriction_{\mathcal L}$.