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Given a cardinal number $n$, an n-ary operation on a set $S$ is a function $\prod_{i:[n]} (-)_i \colon S^n \to S$ from the cartesian power $S^n$ to $S$, where [n] is a set with $n$ elements. The arity of the operation is $n$.
How are we to read $\prod_{i:[n]} (-)_i \colon S^n \to S$? Am I being slow?
Also, we need ’arity’ for relations. E.g., at signature (in logic) we have
A set $Rel(\Sigma)$ whose elements are called relation symbols, equipped with a function $ar: Rel(\Sigma) \to S^*$ to the free monoid on $S$ which prescribes an arity for each relation symbol,
How are we to read $\prod_{i:[n]} (-)_i \colon S^n \to S$? Am I being slow?
I am also confused. An n-ary operation on a set $S$ is just a function $S^n \to S$.
I guess it’s just a typo coming from a change of mind between writing $S^n$ or $\underset{i \in [n]}{\prod} S$.
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