Definition:Reduct of Structure

Definition

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

Let $\LL$ be a subsignature of $\LL'$.

Let $\AA, \AA'$ be structures for $\LL, \LL'$, respectively.

Then $\AA$ is called the reduct of $\AA'$ to $\LL$ if and only if:

For all function symbols $f$ of $\LL$, one has $f_{\AA'} = f_\AA$

For all predicate symbols $p$ of $\LL$, one has $p_{\AA'} = p_\AA$

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

Symbolically, one may write $\AA = \AA' {\restriction_\LL}$.

Also see

• Definition:Expansion of Structure

Sources

• 2009: Kenneth Kunen: The Foundations of Mathematics ... (previous) ... (next): $\mathrm{II}.8$ Further Semantic Notions: Definition $\mathrm{II.8.14}$