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The empty set is an $n$-ary relation for every $n$, right?

How should we call a pair $(n;r)$ consisting of some number $n$ and an $n$-ary relation $r$?

To specify $n$ is necessary only when $r$ is empty, but because there are no reason for $r$ not to be empty, I need to specify $n$ explicitly.

Any term describing this situation?

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    The empty set is a relatio$n$ if and only if your definition of a relation is one that the empty set satisfies vacuously.2012-03-26

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I don't believe that there is any common terminology for what you are asking about, but you should be careful.

Using the common recursive/inductive definition of $n$-tuples, where $\langle a_0 , \ldots , a_{n-1} , a_n \rangle = \langle \langle a_0 , \ldots , a_{n-1} \rangle , a_n \rangle,$ it follows that an $n$-tuple is also an $m$-tuple for all $m \leq n$. Therefore an $n$-ary relation is techincally also an $m$-ary relation for all $m \leq n$. (Differences may arise when talking about $n$-ary relations on a specific set $X$, but even here there might be ambiguity.)

Added due to comments below: Ignore the second paragraph.

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    @Asaf: Yeah, I know that there are various methods of handling the formal definition. I just wanted to be certain that circularity wasn't hanging around.2012-03-26