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So I was just given the definition of a signature:

A signature is a pair $\Sigma = (\Omega, \Pi)$ where $\Omega$ is a set of operation symbols $\omega$, each equipped with an arity $\alpha(\omega)\in\mathbb{N}$ and $\Pi$ is a set of relation symbols $\pi$, each equipped with an arity $\alpha(\pi)\in\mathbb{N}$.

And I saw the term "first order signature" after this without any previous context. Can anyone tell me what is a "first order signature"? Thanks a lot.

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    Probably "first-order" is used there to emphasize that the operations and relations are not higher-order, i.e. they are defined on powers of the carrier, not on powersets (and iterations thereof), i.e. they operate on elements of the carrier, not subsets.2012-11-02

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