The basic cascading arrow notation introduces four new symbols: "→", "·", and "()".
The symbol "↑" (up-arrow) stands for the hyper operator cascader.
The symbol "→" (right-arrow) stands for the cascader power (also called cascadent or powercascadent).
The symbol "·" (middle dot) stands for the product of the arrows (also called arrow-product).
The symbols "()" (parentheses) stands for the order of operations.
Arrow operators of the form ↑→X·↑→X·↑→X·...·↑→X·↑→X·↑→X are known as an arrow-product of cascaders, and each ↑→X is a single cascader. X can be any positive integer or itself can be an arrow-product of cascaders. This recursive definition defines all possible arrows in the cascading arrow notation. Consequently, even arrows in the form ↑↑↑↑↑...↑↑↑↑↑ with n copies of arrows, where n is a positive integer, are also considered a arrow-product of cascaders.
The key band is a cascader of the form ↑↑↑↑↑...↑↑↑↑↑ with n copies of arrows, where n is a positive integer, situated at the right most position of the arrow-product.
The following search-algorithm finds and defines the key band:
Begin at ground level (level 0), and proceed to step 2.
For the current level, find the last cascader of the current hyper-product, and proceed to step 3.
If this cascader is in the form ↑↑↑↑↑...↑↑↑↑↑ with n copies of arrows, then it is the key band by definition, otherwise proceed to step 4.
Go up to the next exponent level of the last cascader and go back to step 2.
Let 'a' to be positive integers, and 'b' to be non-negative integers (positive integers and zero), and @ to be the latter of the expression.
a ↑ b = a^b (a to the power of b).
a ↑^c 1 = a (where c is a positive integer)
a ↑^(c + 1) (b + 1) = a ↑^c (a ↑^(c+1) b)
If the last cascader is not in the form ↑↑↑↑↑...↑↑↑↑↑ with n copies of arrows, a @·X↑→(X↑^(n + 1)) b = a @·X↑→(X↑^(n))^b a
If the right-hand side argument is 1, and the last cascader is in the form ↑↑↑↑↑...↑↑↑↑↑ with n copies of arrows, it can be degenerated: a @·↑^n 1 = a.
Otherwise, if the right-hand side argument is greater than 1, and the last cascader is in the form ↑↑↑↑↑...↑↑↑↑↑ with n copies of arrows: a @·X↑^(n + 1) (b + 1) = a @·X↑^n (a @·X↑^(n + 1) b).
3 ↑→↑ 3 = 3 ↑^3 3 = 3 ↑↑↑ 3 = 3 ↑↑ (3 ↑↑↑ 2) = 3 ↑↑ (3 ↑↑ (3 ↑↑↑ 1)) = 3 ↑↑ (3 ↑↑ 3) = 3 ↑↑ (3 ↑ (3 ↑↑ 2)) = 3 ↑↑ (3 ↑ (3 ↑ (3 ↑↑ 1))) = 3 ↑↑ (3 ↑ 3 ↑ 3) = 3 ↑↑ (3 ↑ 27) = 3 ↑↑ 7,625,597,484,987 = tritri
10 ↑→↑ 10 = 10 ↑^10 10 = 10 ↑↑↑↑↑↑↑↑↑↑ 10 = tridecal
4 ↑→↑·↑ 3 = 4 ↑→↑ (4 ↑→↑·↑ 2) = 4 ↑→↑ 4 ↑→↑ 4 = 4 ↑→↑ 4 ↑^4 4 = 4 ↑→↑ 4 ↑↑↑↑ 4 = 4 ↑^(4 ↑↑↑↑ 4) 4
3 ↑→↑·↑ 65 = 3 ↑→↑ (3 ↑→↑·↑ 64) = 3 ↑→↑ (3 ↑→↑ (3 ↑→↑·↑ 63)) = ... ≈ Graham's number
10 ↑→↑·↑→↑ 100 = 10 ↑→↑·↑^10 10 = 10 ↑→↑·↑↑↑↑↑↑↑↑↑↑ 10 = biggol
5 ↑→↑↑ 3 = 5 (↑→↑)^3 5 = 5 ↑→↑·↑→↑·↑→↑ 5 = 5 ↑→↑·↑→↑·↑^5 5 = 5 ↑→↑·↑→↑·↑↑↑↑↑ 5
4 ↑→↑→↑ 4 = 4 ↑→↑^4 4 = 4 ↑→↑↑↑↑ 4 = 4 (↑→↑↑↑)^4 4 = 4 ↑→↑↑↑·↑→↑↑↑·↑→↑↑↑·↑→↑↑↑ 4 = 4 ↑→↑↑↑·↑→↑↑↑·↑→↑↑↑·(↑→↑↑)^4 4 = 4 ↑→↑↑↑·↑→↑↑↑·↑→↑↑↑·↑→↑↑·↑→↑↑·↑→↑↑·↑→↑↑ 4 = 4 ↑→↑↑↑·↑→↑↑↑·↑→↑↑↑·↑→↑↑·↑→↑↑·↑→↑↑·(↑→↑)^4 4 = 4 ↑→↑↑↑·↑→↑↑↑·↑→↑↑↑·↑→↑↑·↑→↑↑·↑→↑↑·↑→↑·↑→↑·↑→↑·↑→↑ 4 = 4 ↑→↑↑↑·↑→↑↑↑·↑→↑↑↑·↑→↑↑·↑→↑↑·↑→↑↑·↑→↑·↑→↑·↑→↑·↑^4 4 = 4 ↑→↑↑↑·↑→↑↑↑·↑→↑↑↑·↑→↑↑·↑→↑↑·↑→↑↑·↑→↑·↑→↑·↑→↑·↑↑↑↑ 4
3 ↑→↑→↑↑↑ 2 = 3 ↑→(↑→↑↑)^2 3 = 3 ↑→(↑→↑↑·↑→↑↑) 3 = 3 ↑→(↑→↑↑·(↑→↑)^3) 3 = 3 ↑→(↑→↑↑·↑→↑·↑→↑·↑→↑) 3 = 3 ↑→(↑→↑↑·↑→↑·↑→↑·↑^3) 3 = 3 ↑→(↑→↑↑·↑→↑·↑→↑·↑↑↑) 3 = 3 (↑→(↑→↑↑·↑→↑·↑→↑·↑↑))^3 3 = 3 ↑→(↑→↑↑·↑→↑·↑→↑·↑↑)·↑→(↑→↑↑·↑→↑·↑→↑·↑↑)·↑→(↑→↑↑·↑→↑·↑→↑·↑↑) 3 = ...
6 ↑→↑→↑→↑→↑ 3 = 6 ↑→↑→↑→↑^3 6 = 6 ↑→↑→↑→↑↑↑ 6 = 6 ↑→↑→(↑→↑↑↑)^6 6 = 6 ↑→↑→(↑→↑↑·↑→↑↑·↑→↑↑·↑→↑↑·↑→↑↑·↑→↑↑) 6 = ...
That's an extension of the Knuth's up-arrow notation. Instead of a minor awkwardness in BEAF and Bird's array notation that Aeton's N-growing hierarchy does not match exactly for ordinal levels α ≥ ω^ω level while the cascading arrow notation does.
The basic cascading arrow notation allows to obtain the most contingent increment to reach the level comparable to ε0 with respect to Wainer's hierarchy in the fast-growing hierarchy. Using FGH in comparison, single up-arrow symbol corresponds to 1 (or ω^0), the arrow-product symbol corresponds to the ordinal addition, and the cascade function ↑→ corresponds to the omega-exponentiation.
I also proposed the extension that diagonalizes the limit of the notation level: the ↑[↓]↑ arrow, which is defined as a ↑[↓]↑ b = a ↑→↑→↑→...→↑→↑→↑ a with b copies of ↑'s in the cascadent; and continues to the next level of the notation, called "extended cascading arrow notation". Example: 4 ↑[↓]↑ 4 = 4 ↑→↑→↑→↑ 4 = 4 ↑→↑→↑↑↑↑ 4 = 4 ↑→(↑→↑↑↑·↑→↑↑↑·↑→↑↑↑·↑→↑↑↑) 4