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Let $A$ be some space and $N$ be an integer. Does a function of the form $f:A^N\to A$ have a particular name? I'm looking for a keyword to see if there are any general properties of functions of this type.

jonem
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  • That depends, is $A$ a space of scalars, or any space? – R. Burton Jan 08 '19 at 16:48
  • Usually it's $f:N\rightarrow A^N \rightarrow A$ and it's called evaluation, i.e. normal function call in any programming language. – tp1 Jan 08 '19 at 16:50
  • @R.Burton Any space will do – jonem Jan 08 '19 at 16:50
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    $N$-ary operator? (See https://en.wikipedia.org/wiki/Arity.) –  Jan 08 '19 at 17:03
  • @jonem I'm not sure then. If $A$ is a scalar space, then $f$ belongs to the class of scalar functions (http://mathworld.wolfram.com/ScalarFunction.html). But if $f$ is tensor-valued, then it could be almost anything. You could consider $f$ to be the class of all possible index/dimension reductions (of which tensor contraction would be a special case). But you would have to allow the Cartesian product of two $n$-dimensional objects to yield an $n^2$-dimensional object (i.e. the vector $((x_1,x_2),(y_1,y_2))$ is equivalent to the matrix $\left(\begin{matrix}x_1&x_2\y_1&y_2\end{matrix}\right)$). – R. Burton Jan 08 '19 at 17:03

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$f$ is an operation on the set $A$ of arity $N.$ More specifically, a finitary operation.

Of course "operation" has many meanings, but this meaning is certainly common particularly when discussing algebraic theories in general. For example it is used in Grätzer's Universal Algebra, a standard reference.

Dap
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