# Is there a mathematical equivalent to a variadic function?

I'm looking to express an operation on collections of elements from a set. In programming, this would more or less be a variadic function. Is there an equivalent in mathematics?

• Perhaps a function on the power set of a set? – Michael Burr Mar 20 '16 at 1:29
• That could work, but how would I represent the stop of the collection? So let's just go with a finite version with a maximum of 4 elements in the set. How would I designate a collection of two elements? – Daniel Goldman Mar 20 '16 at 1:30
• Are repeats allows? Does order matter? – Michael Burr Mar 20 '16 at 1:30
• Order matters and repeats are allowed. – Daniel Goldman Mar 20 '16 at 1:31
• There aren't pleasant ways to write this, but the set of interest is $\bigcup_{i=0}^\infty A^i$ where $A^i$ is the Cartesian product of your set $A$ with itself $i$ times. See this question. Each element of this set represents an ordered list of finite length. You're talking about a function on this set. – Michael Burr Mar 20 '16 at 1:33

Let $A=\{1,2,3,4\}$. Then:

• $A^0=\{\emptyset\}$ (this definition is somewhat nonstandard, but helps for your application)

• $A^1=\{1,2,3,4\}$.

• $A^2=\{(1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(2,3),(2,4),(3,1),(3,2),(3,3),(3,4),(4,1),(4,2),(4,3),(4,4)\}$.

The sets continue, but get larger and I don't want to write them.

Each element of $\bigcup_{i=0}^\infty A^i$ is an ordered list of elements (throw all the elements of all the different lengths together into a single collection). You don't need a "stop" because the length is part of the element.

• This should work just fine, and would extend to uncountable indexes in a reasonable way. I'll go with that. Thanks. – Daniel Goldman Mar 20 '16 at 1:41
• I'm still kind of amazed there is no name for this structure. – Daniel Goldman Mar 20 '16 at 4:21