Extended arrow notation is a notation that was invented by Googology Wikia User Googleaarex: <a href="https://googology.wikia.com/wiki/User:Googleaarex/" class="external autonumber">[1]</a>. # Basic Notation Basic Notation is very simple. It generalizes the [](Knuth's_up-arrow_notation.md). \(a \uparrow_2 b = a \underbrace{\uparrow\uparrow\dots\uparrow}_b a\) \(a \uparrow_3 b = a \underbrace{\uparrow_2\uparrow_2\dots\uparrow_2}_b a\) \(a \uparrow_n b = a \underbrace{\uparrow_{n-1}\uparrow_{n-1}\dots\uparrow_{n-1}}_b a\) Note that all parts of Extended arrow notation, like Knuth's up-arrow notation, have expressions that are evaluated from the right. Limit in FGH: \(f_\omega(n)\) # Nested up-arrow notation To extend the notation here, we first have to make a change: \(\uparrow_n = \uparrow_{\underbrace{\uparrow\uparrow\dots\uparrow}_{n-1}}\) Then we turn the problem into Basic notation: \(a \uparrow_{\uparrow_2} b = a \uparrow_{\underbrace{\uparrow\uparrow\dots\uparrow}_b} a = a \uparrow_{b+1} b\), and \(a \uparrow_{\uparrow\uparrow_2} b = a \underbrace{\uparrow_{\uparrow_2}\uparrow_{\uparrow_2}\dots\uparrow_{\uparrow_2}}_b a\) Then: \(a \uparrow_{\uparrow_{\uparrow_2}} b = a \uparrow_{\uparrow_{b+1}} a\) and so on. Limit: \(\varepsilon_0\) # Array up-arrow notation ## \(\Omega\) typed arrows Limit: \(\psi(\varepsilon_{\Omega+1})\) ## \(\Omega_2\) typed arrows Limit: \(\psi(\psi_1(\varepsilon_{\Omega_2+1}))\) ## \(\Omega_3\) typed arrows and beyond Limit: \(\psi(\psi_I(0))\) ## Inaccesible arrows Limit: \(\psi(\psi_{I(1,0)}(0))\) ## 1-inaccesible arrows and beyond Limit: \(\psi(\psi_{I(\omega, 0)}(0))\) # Dimensional array up-arrow notation Limit: \(\psi(\psi_{\chi(\varepsilon_{M+1})}(0))\) # Hyperarray up-arrow notation Limit: \(\psi(\psi_{\chi(M(1,0))}(0))\) # Legion array up-arrow notation ## Layered arrays Limit: \(\psi(\psi_{ {\Xi(1)}^\omega}(0))\) ## The hyperseparator Limit: \(\psi(\psi_{M(1,\Xi(1)+1)}(0))\) ## The second hyperseparator Limit:??? # Hyperlegion array up-arrow notation