User:Googleaarex/Extended Up-Arrow Notation/Basic

At Up-Arrow Notation, this one has growth rate \(\omega\)!

I will extend this notation. This one called Extended Up-Arrow Notation. I trying to reach \(\omega^\omega\).

2-Typed Arrows
Let 2nd type of up-arrow is down-arrow.

\(a \downarrow b = a \uparrow^b a\) (growth rate \(\omega\))

\(a \downarrow\uparrow b = \underbrace{a \downarrow a ... a \downarrow a}_{b}\) (growth rate \(\omega + 1\))

\(a \square\uparrow b = \underbrace{a \square a ... a \square a}_{b}\) (growth rate \(\square + 1\))

\(\square\) is number of up-arrows and down-arrows.

\(a \square\downarrow b = a \square\uparrow^b a\) (growth rate \(\square + \omega\)).

The limit growth rate is \(\omega^2\)!

3-Typed Arrows
Let 3rd type of up-arrow is left-arrow.

\(a \leftarrow b = ???\)