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

Then $\mathcal L$ is said to be a subsignature of $\mathcal L'$, denoted $\mathcal L \subseteq \mathcal L'$, if and only if, for each $n \in \N$:

$\mathcal F_n \left({\mathcal L}\right) \subseteq \mathcal F_n \left({\mathcal L'}\right)$
$\mathcal P_n \left({\mathcal L}\right) \subseteq \mathcal P_n \left({\mathcal L'}\right)$

where $\mathcal F_n$ denotes the collection of $n$-ary function symbols, and $\mathcal P_n$ denotes the collection of $n$-ary predicate symbols.


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

Then $\mathcal L'$ is said to be a supersignature of $\mathcal L$, denoted:

$\mathcal L' \supseteq \mathcal L$

Also see