Proposal for a symbol for hyperoperation aggregation [closed]

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Mathematicians use ∑ for repeated addition and ∏ for repeated multiplication.
I’ve been exploring whether we can generalize this pattern for higher hyperoperations — such as exponentiation, tetration, etc.

Proposed notation

I propose a "SmileyFold" operator (☺) that represents folding a finite sequence of terms under an arbitrary hyperoperation level n.

For example:

• ☺1^{1→3}(3x−1) = 2 + 5 + 8
• ☺2^{1→3}(3x−1) = ×ばつ8
• ☺3^{1→3}(3x−1) = 2^{5^{8}}

Formally, for a function f defined on {a, a+1, ..., b}: Sn(x1) = x1
Sn(x1, x2, ..., xk) = x1 [n] Sn(x2, ..., xk)

where [n] denotes the nth hyperoperation (addition, multiplication, exponentiation, etc.).

Question

Is there an established or equivalent notation for this kind of generalization?
If not, does this definition and notation seem consistent or useful?

• user619894: have you considered the for loop?

Answer 1

There are no notations specifically for this, but we generally use 'big operators' when we want to represent what programmers would call 'folds'. So if we have... $$a_1 \oplus a_2 \oplus a_3 \oplus a_4$$

...we might write it as $$\bigoplus_{i=1}^4 a_i\qquad\text{or even just}\qquad \bigoplus a\text{.}$$

But no, this does not seem useful, unfortunately. This is because the higher hyperoperations - beyond exponentiation - don't actually appear in math.

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We care about addition and multiplication in part because they are two basic operations that have a lot of nice properties: they're commutative and associative, they both have identities, and the distributive law 'links' them together. And of course, both of them are operations we often do in real life They're an obvious and natural thing to want to study.

Exponentiation appears much less often - still often, of course, but much less. It typically comes from either repeated multiplication (which you can do in any ring), or the exponential function ($x\mapsto e^x$). Either way, the base and exponent are playing fundamentally different 'roles'. And so, repeated exponentiation of the same number doesn't pop up all the time, which means it's not interesting to most mathematicians.