Is there a generalized analogue to the summation and product operators?

Question

The summation operator ($\sum$) and the product operator ($\prod$) will take a sequence of values and sum or multiply them.

Is there a general operator that will take a sequence of values and repeatedly apply a binary function to them?

Something, such that:

$\xi_{n=0}^5 S_n = f(S_5,\xi_{n=0}^4S_n)$

$\xi_{n=0}^5 S_n = f(S_5,f(S_4, \xi_{n=0}^3S_n))$

$...$

$\xi_{n=0}^5 S_n = f(S_5,f(S_4,f(S_3,f(S_2,f(S_1,f(S_0, I_f))))))$

Where $I_f$ is the identity value for the operation in question.

Answer 1

It seems I was looking for an iterated binary operation which is denoted by

$f/(a_0, a_1, a_2, ..., a_n)$

Answer 2

You need an associative binary operation for unambiguous meaning: otherwise for each way of inserting parenthesis one will get a different answer. If the individual entities are sets one can consider union, and intersections for them.

Also one needs a convention for what is the value for empty operation (that is, when the set of terms being 'summed' or 'multiplied' is empty).

In numbers one can talk of gcd of n numbers, lcm of n numbers.

So in any semigroup one can talk of this.