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 $\AA \subseteq \BB$, :


 * $A \subseteq B$
 * 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 $p_\AA = p_\BB$