Definition:Substructure

Definition

Let $\AA, \BB$ be structures for a signature $\LL$.

Let $A, B$ be their respective underlying sets.

Then $\AA$ is a substructure of $\BB$, denoted $\AA \subseteq \BB$, if and only if:

$A \subseteq B$

For each function symbol $f$ of arity $n$, we have $f_\AA = f_\BB \restriction_{A^n}$, where $\restriction$ denotes restriction

For each predicate symbol $p$ of arity $n$, we have $p_\AA = p_\BB \restriction_{A^n}$

Note that in particular, for $n = 0$, this reduces to $f_\AA = f_\BB$ and $p_\AA = p_\BB$

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Sources

• 2009: Kenneth Kunen: The Foundations of Mathematics ... (previous) ... (next): $\text{II}.8$ Further Semantic Notions: Definition $\text{II}.8.17$