FANDOM

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Well I had to do it at some point.

Linear arrays - up to $$\omega^\omega$$

$$b(a,a)$$

$$a^2$$
$$b(a,a,1)$$ $$f_2(a)$$
$$b(a,a,n)$$ $$f_{n+1}(a)$$
$$b(a,a,0,1)$$ $$f_{\omega}(a)$$
$$b(a,a,n,1)$$ $$f_{\omega +n}(a)$$
$$b(a,a,n,k)$$ $$f_{\omega k + n}(a)$$
$$b(a,a,0,0,1)$$ $$f_{\omega^2}(a)$$
$$b(a,a,n,k,i)$$ $$f_{\omega^2 i + \omega k + n}(a)$$
$$b(a,a,0,0,0,1)$$ $$f_{\omega^3}(a)$$
$$b(a,a,n,k,l,m)$$ $$f_{\omega^3 m + \omega^2 l + \omega k + n}(a)$$
$$b(a,a,b,c,d,e,.....)$$ $$f_{...... + \omega^3 e + \omega^2 d + \omega c +b}(a)$$

Limit of l-ABHAN is $$\omega^\omega$$.

Two rows - up to $$\omega^{\omega 2}$$

$$b(a,a\{1\}1)$$ $$f_{\omega^\omega}(a)$$
$$b(a,a,1\{1\}1)$$ $$f_{\omega^\omega +1}(a)$$
$$b(a,a,n\{1\}1)$$ $$f_{\omega^\omega +n}(a)$$
$$b(a,a,0,1\{1\}1)$$ $$f_{\omega^\omega + \omega}(a)$$
$$b(a,a,n,k\{1\}1)$$ $$f_{\omega^\omega + \omega k + n}(a)$$
$$b(a,a,b,c,d,.....\{1\}1)$$ $$f_{\omega^\omega + ..... +\omega^2 d + \omega c + b}(a)$$
$$b(a,a\{1\}2)$$ $$f_{\omega^\omega 2}(a)$$
$$b(a,a,b,c,d,e,....\{1\}n)$$                                  $$f_{\omega^\omega n + ...... \omega^3 e + \omega^2 d + \omega c +b}(a)$$
$$b(a,a\{1\}0,1)$$ $$f_{\omega^{\omega +1}}(a)$$
$$b(a,a,b,c,d,e,...\{1\}n,1)$$                    $$f_{\omega^{\omega +1} + \omega^\omega n + ...... \omega^3 e + \omega^2 d + \omega c +b}(a)$$
$$b(a,a\{1\}0,2)$$ $$f_{\omega^{\omega +1} 2}(a)$$
$$b(a,a\{1\}0,0,1)$$ $$f_{\omega^{\omega +2}}(a)$$
$$b(a,a\{1\}n_1,n_2,.....,n_i,n_{i+1})$$ $$f_{\omega^{\omega + i}n_{i+1} + \omega^{\omega + i - 1}n_{i} + ...... + \omega^{\omega +1}n_2 + \omega^{\omega}n_1}(a)$$

Planar - up to $$\omega^{\omega^2}$$

$$b(a,a\{1\}0\{1\}1)$$ $$f_{\omega^{\omega 2}}(a)$$
$$b(a,a,1\{1\}0\{1\}1)$$ $$f_{\omega^{\omega 2} +1}(a)$$
$$b(a,a,b,c,d,e....\{1\}0\{1\}1)$$ $$f_{\omega^{\omega 2} +.....+\omega^3 e \omega^2 d + \omega c + b}(a)$$
$$b(a,a\{1\}1\{1\}1)$$ $$f_{\omega^{\omega 2} + \omega^\omega}(a)$$
$$b(a,a\{1\}b,c,d,e,....\{1\}1)$$                          $$f_{\omega^{\omega 2} + ..... + \omega^{\omega +3}e +\omega^{\omega +2}d + \omega^{\omega +1}c + \omega^\omega b}$$
$$b(a,a\{1\}0\{1\}2)$$ $$f_{\omega^{\omega 2} 2}(a)$$
$$b(a,a\{1\}0\{1\}b,c,d,e,...)$$ $$f_{..... + \omega^{\omega 2 + 3}e + \omega^{\omega 2 + 2}d + \omega^{\omega 2 + 1}c + \omega^{\omega 2}b}$$
$$b(a,a\{1\}0\{1\}0\{1\}1)$$ $$f_{\omega^{\omega 3}}(a)$$
$$b(a,a\{1\}n_1\{1\}n_2,......\{1\}n_i)$$ $$f_{\omega^{\omega i}n_i + ..... + \omega^{\omega 2}n_2 + \omega^{\omega}n_1}(a)$$
$$b(a,a\{1\}0\{1\}0\{1\}0\{1\}.....\{1\}0\{1\}1)$$ w/ $$n$$ $$\{1\}$$'s

$$f_{\omega^{\omega n}}(a)$$

Cubic - up to $$\omega^{\omega^3}$$

$$b(a,a\{2\}1)$$

$$f_{\omega^{\omega^2}}(a)$$
$$b(a,a,1\{2\}1)$$ $$f_{\omega^{\omega^2}+1}(a)$$
$$b(a,a,0,1\{2\}1)$$ $$f_{\omega^{\omega^2}+\omega}(a)$$
$$b(a,a\{1\}0\{2\}1)$$ $$f_{\omega^{\omega^2} + \omega^\omega}(a)$$
$$b(a,a\{1\}1\{2\}1)$$ $$f_{\omega^{\omega^2}+\omega^\omega 2}(a)$$
$$b(a,a\{1\}0,1\{2\}1)$$ $$f_{\omega^{\omega^2}+\omega^{\omega +1}}(a)$$
$$b(a,a\{1\}0\{1\}0\{2\}1)$$ $$f_{\omega^{\omega^2}+\omega^{\omega 2}}(a)$$
$$b(a,a\{2\}2)$$ $$f_{\omega^{\omega^2}2}(a)$$
$$b(a,a\{2\}0,1)$$ $$f_{\omega^{\omega^2 +1}}(a)$$
$$b(a,a\{2\}0\{1\}1)$$ $$f_{\omega^{\omega^2 + \omega}}(a)$$
$$b(a,a\{2\}0\{1\}0\{1\}1)$$ $$f_{\omega^{\omega^2 + \omega 2}}$$
$$b(a,a\{2\}0\{2\}1)$$ $$f_{\omega^{\omega^2 2}}(a)$$
$$b(a,a\{2\}0\{2\}0\{2\}1)$$ $$f_{\omega^{\omega^2 3}}(a)$$

$$b(a,a\{2\}0\{2\}0\{2\}.....\{2\}1)$$ w/ $$n$$ $$\{2\}$$'s

$$f_{\omega^{\omega^2 n}}(a)$$

Multi-dimentional - up to $$\omega^{\omega^\omega}$$

$$b(a,a\{3\}1)$$ $$f_{\omega^{\omega^3}}(a)$$
$$b(a,a,1\{3\}1)$$ $$f_{\omega^{\omega^3}+1}(a)$$
$$b(a,a,0,1\{3\}1)$$ $$f_{\omega^{\omega^3}+\omega}(a)$$
$$b(a,a\{1\}0\{3\}1)$$ $$f_{\omega^{\omega^3}+\omega^\omega}(a)$$
$$b(a,a\{2\}0\{3\}1)$$ $$f_{\omega^{\omega^3}+\omega^{\omega^2}}(a)$$
$$b(a,a\{3\}2)$$ $$f_{\omega^{\omega^3}2}(a)$$
$$b(a,a\{3\}0,1)$$ $$f_{\omega^{\omega^3 +1}}$$
$$b(a,a\{3\}0\{1\}1)$$ $$f_{\omega^{\omega^3 +\omega}}(a)$$
$$b(a,a\{3\}0\{3\}1)$$ $$f_{\omega^{\omega^3 2}}(a)$$
$$b(a,a\{4\}1)$$ $$f_{\omega^{\omega^4}}(a)$$
$$b(a,a\{4\}0,1)$$ $$f_{\omega^{\omega^4 +1}}(a)$$
$$b(a,a\{4\}0\{4\}1)$$ $$f_{\omega^{\omega^4 2}}(a)$$
$$b(a,a\{4\}0\{4\}0\{4\}......\{4\}1)$$ w/ $$n$$ $$\{4\}$$'s $$f_{\omega^{\omega^4 n}}(a)$$
$$b(a,a{5}1)$$ $$f_{\omega^{\omega^5}}(a)$$
$$b(a,a\{n\}1)$$ $$f_{\omega^{\omega^n}}(a)$$

Limit of md-ABHAN is $$\omega^{\omega^\omega}$$

Hyperdimentional - up to $$\varepsilon_0$$

$$b(a,a\{0,1\}1)$$ $$f_{\omega^{\omega^\omega}}(a)$$
$$b(a,a,1\{0,1\}1)$$ $$f_{\omega^{\omega^\omega}+1}(a)$$
$$b(a,a\{1\}0\{0,1\}1)$$ $$f_{\omega^{\omega^\omega}+\omega^\omega}(a)$$
$$b(a,a\{n\}0\{0,1\}1)$$ $$f_{\omega^{\omega^\omega}+\omega^{\omega^n}}(a)$$
$$b(a,a\{0,1\}2)$$ $$f_{\omega^{\omega^\omega}2}(a)$$
$$b(a,a\{0,1\}0,1)$$ $$f_{\omega^{\omega^\omega +1}}(a)$$
$$b(a,a\{0,1\}0\{1\}1)$$ $$f_{\omega^{\omega^\omega +\omega}}(a)$$
$$b(a,a\{0,1\}0\{0,1\}1)$$ $$f_{\omega^{\omega^\omega 2}}(a)$$
$$b(a,a\{1,1\}1)$$ $$f_{\omega^{\omega^{\omega +1}}}(a)$$
$$b(a,a\{n,1\}1)$$ $$f_{\omega^{\omega^{\omega +n}}}(a)$$
$$b(a,a\{0,2\}1)$$ $$f_{\omega^{\omega^{\omega 2}}}(a)$$
$$b(a,a\{b,c,d,e,....\}1)$$ $$f_{\omega^{\omega^{.... +\omega^3 e + \omega^2 d + \omega c +b}}}(a)$$
$$b(a,a\{0\{1\}1\}1)$$ $$f_{\omega^{\omega^{\omega^\omega}}}(a)$$
$$b(a,a\{0\{n\}1\}1)$$ $$f_{\omega^{\omega^{\omega^{\omega^n}}}}(a)$$
$$b(a,a\{0\{b,c,d,e,...\}1\}1)$$ $$f_{\omega^{\omega^{\omega^{\omega^{.....+\omega^3 e + \omega^2 d + \omega c +b}}}}}(a)$$
$$b(a,a\{0\{0\{1\}1\}1\}1)$$ $$f_{\omega^{\omega^{\omega^{\omega^{\omega^\omega}}}}}(a)$$
$$b(a,a\{0\{0\{b,c,d,e,...\}1\}1\}1)$$ $$f_{\omega^{\omega^{\omega^{\omega^{\omega^{.....+\omega^3 e + \omega^2 d + \omega c +b}}}}}}(a)$$
$$b(a,a\{0\{0\{0\{1\}1\}1\}1\}1)$$ $$f_{\omega^{\omega^{\omega^{\omega^{\omega^{\omega^\omega}}}}}}(a)$$
$$b(a,a\{0\{0\{0\{0,1\}1\}1\}1\}1)$$ $$f_{\omega^{\omega^{\omega^{\omega^{\omega^{\omega^{\omega^\omega}}}}}}}(a)$$
$$b(a,a\{0\{0\{0\{..\{1\}..\}1\}1\}1\}1)$$ w/ $$n$$ ones from the center out.       $$f_{\omega\uparrow\uparrow (n*2)}(a)$$

Limit of hd-ABHAN is $$\varepsilon_0$$.

Nested (only linear) - up to $$\varphi(\omega,0)$$

Up to $$\varepsilon_1$$

$$b(a,a\{0\backslash 1\}1)$$ $$f_{\varepsilon_0}(a)$$
$$b(a,a,1\{0\backslash 1\}1)$$ $$f_{\varepsilon_0 +1}(a)$$
$$b(a,a,0,1\{0\backslash 1\}1)$$ $$f_{\varepsilon_0 +\omega}(a)$$
$$b(a,a\{1\}0\{0\backslash 1\}1)$$ $$f_{\varepsilon_0 +\omega^\omega}(a)$$
$$b(a,a\{0,1\}0\{0\backslash 1\}1)$$ $$f_{\varepsilon_0 +\omega^{\omega^\omega}}(a)$$
$$b(a,a\{0\{1\}1\}0\{0\backslash 1\}1)$$ $$f_{\varepsilon_0 +\omega^{\omega^{\omega^\omega}}}(a)$$
$$b(a,a\{0\{0,1\}1\}0\{0\backslash 1\}1)$$ $$f_{\varepsilon_0 +\omega^{\omega^{\omega^{\omega^\omega}}}}(a)$$
$$b(a,a\{0\backslash 1\}2)$$ $$f_{\varepsilon_0 2}(a)$$
$$b(a,a\{0\backslash 1\}0,1)$$ $$f_{\varepsilon_0 \omega}(a)$$
$$b(a,a\{0\backslash 1\}0\{1\}1)$$ $$f_{\varepsilon_0 \omega^\omega}(a)$$
$$b(a,a\{0\backslash 1\}0\{0,1\}1)$$ $$f_{\varepsilon_0 \omega^{\omega^\omega}}(a)$$
$$b(a,a\{0\backslash 1\}0\{0\backslash 1\}1)$$ $$f_{\varepsilon_0^2}(a)$$
$$b(a,a\{0\backslash 1\}0\{0\backslash 1\}0\{0\backslash 1\}1)$$ $$f_{\varepsilon_0^3}(a)$$
$$b(a,a\{1\backslash 1\}1)$$ $$f_{\varepsilon_0^\omega}(a)$$
$$b(a,a\{1\backslash 1\}n)$$ $$f_{\varepsilon_0^\omega n}(a)$$
$$b(a,a\{1\backslash 1\}0\{1\backslash 1\}1)$$ $$f_{\varepsilon_0^{\omega 2}}(a)$$
$$b(a,a\{2\backslash 1\}1)$$ $$f_{\varepsilon_0^{\omega^2}}(a)$$
$$b(a,a\{n\backslash 1\}1)$$ $$f_{\varepsilon_0^{\omega^n}}(a)$$
$$b(a,a\{0,1\backslash 1\}1)$$ $$f_{\varepsilon_0^{\omega^\omega}}(a)$$
$$b(a,a\{0\{1\}1\backslash 1\}1)$$ $$f_{\varepsilon_0^{\omega^{\omega^\omega}}}(a)$$
$$b(a,a\{0\{0,1\}1\backslash 1\}1)$$ $$f_{\varepsilon_0^{\omega^{\omega^{\omega^\omega}}}}(a)$$
$$b(a,a\{0\{0\backslash1\}1\backslash 1\}1)$$ $$f_{\varepsilon_0^{\varepsilon_0}}(a)$$
$$b(a,a\{0\{0\{0\backslash 1\}1\backslash 1\}1\backslash 1\}1)$$ $$f_{\varepsilon_0^{\varepsilon_0^{\varepsilon_0}}}(a)$$
$$b(a,a\{0\{0\{0\{0\backslash 1\}1\backslash 1\}1\backslash 1\}1\backslash 1\}1)$$ $$f_{\varepsilon_0^{\varepsilon_0^{\varepsilon_0^{\varepsilon_0}}}}(a)$$
$$b(a,a\{0\{0\{.....\}1\backslash 1\}1\backslash 1\}1)$$ \W $$n$$ $$\backslash$$'s $$f_{\varepsilon_0^{\varepsilon_0^{.....}}}(a)$$ \W $$n$$ $$\varepsilon_0$$'s

Up to $$\zeta_0$$

$$b(a,a\{0\backslash 2\}1)$$ $$f_{\varepsilon_1}(a)$$
$$b(a,a,n\{0\backslash 2\}1)$$ $$f_{\varepsilon_1 + n}(a)$$
$$b(a,a,0,1\{0\backslash 2\}1)$$ $$f_{\varepsilon_1 + \omega}(a)$$
$$b(a,a\{1\}0\{0\backslash 2\}1)$$ $$f_{\varepsilon_1 + \omega^\omega}(a)$$
$$b(a,a\{0,1\}0\{0\backslash 2\}1)$$ $$f_{\varepsilon_1 + \omega^{\omega^\omega}}(a)$$
$$b(a,a\{0\{1\}1\}0\{0\backslash 2\}1)$$ $$f_{\varepsilon_1 + \omega^{\omega^{\omega^\omega}}}(a)$$
$$b(a,a\{0\backslash 1\}0\{0\backslash 2\}1)$$ $$f_{\varepsilon_1 + \varepsilon_0}(a)$$
$$b(a,a\{0\backslash 2\}2)$$ $$f_{\varepsilon_1 2}(a)$$
$$b(a,a\{0 \backslash 2\}n)$$ $$f_{\varepsilon_1 n}(a)$$
$$b(a,a\{0\backslash 2\}0,1)$$ $$f_{\varepsilon_1 \omega}(a)$$
$$b(a,a\{0\backslash 1\}0\{1\}1)$$ $$f_{\varepsilon_1 \omega^\omega}(a)$$
$$b(a,a\{0\backslash 2\}0\{0\backslash 1\}1)$$ $$f_{\varepsilon_1 \varepsilon_0}(a)$$
$$b(a,a\{0\backslash 2\}0\{0\backslash 2\}1)$$ $$f_{\varepsilon^{2}_1}(a)$$
$$b(a,n\{1\backslash 2\}1)$$ $$f_{\varepsilon^{n}_1}(a)$$
$$b(a,a\{1\backslash 2\}1)$$ $$f_{\varepsilon^{\omega}_1}(a)$$
$$b(a,a\{0\backslash 2\}0\{1\backslash 2\}1)$$ $$f_{\varepsilon^{\omega}_1 + \varepsilon_1}(a)$$
$$b(a,a\{1\backslash 2\}2)$$ $$f_{\varepsilon^{\omega}_1 2}(a)$$
$$b(a,a\{1\backslash 2\}0\{0\backslash 2\}1)$$ $$f_{\varepsilon^{\omega + 1}_1}(a)$$
$$b(a,a\{1\backslash 2\}0\{1\backslash 2\}1)$$ $$f_{\varepsilon^{\omega 2}}(a)$$
$$b(a,n\{2\backslash 2\}1)$$ $$f_{\varepsilon^{\omega n}_1}$$

From now on,we are going to focus mainly on the seperators.

$$\{2\backslash 2\}$$ $$f_{\varepsilon^{\omega^2}_1}$$
$$\{n\backslash 2\}$$ $$f_{\varepsilon^{\omega^n}_1}$$
$$\{0,1\backslash 2\}$$ $$f_{\varepsilon^{\omega^\omega}_1}$$
$$\{0\{1\}1\backslash 2\}$$ $$f_{\varepsilon^{\omega^{\omega^\omega}}_1}$$
$$\{0\{0,1\}1\backslash 2\}$$ $$f_{\varepsilon^{\omega^{\omega^{\omega^\omega}}}_1}$$
$$\{0\{0\backslash 1\}1\backslash 2\}$$ $$f_{\varepsilon^{\varepsilon_0}_1}$$
$$\{0\{0\{0\backslash 1\}1\backslash 1\}1\backslash 2\}$$ $$f_{\varepsilon^{\varepsilon^{\varepsilon_0}_0}_1}$$
$$\{0\{0\backslash 2\}1\backslash 2\}$$ $$f_{\varepsilon^{\varepsilon_1}_1}$$
$$\{0\backslash 3\}$$ $$f_{\varepsilon_2}$$
$$\{0\backslash n\}$$ $$f_{\varepsilon_{n-1}}$$
$$\{0\backslash 0,1\}$$ $$f_{\varepsilon_{\omega}}$$
$$\{0\backslash 0\{0\backslash 1\}1\}$$ $$f_{\varepsilon_{\varepsilon_0}}$$
$$\{0\backslash 0\{0\backslash 0\{0\backslash 1\}1\}1\}$$ $$f_{\varepsilon_{\varepsilon_{\varepsilon_0}}}$$
$$\{0\backslash 0\{0\backslash 0\{.....\}1\}1\}$$ $$f_{\varepsilon_{\varepsilon_{.....}}}$$

Up to $$\eta_0$$

$$\{0\backslash 0\backslash 1\}$$ $$f_{\zeta_0}$$
$$\{1\backslash 0\backslash 1\}$$ $$f_{\zeta^{\omega}_0}$$
$$\{2\backslash 0\backslash 1\}$$ $$f_{\zeta^{\omega^2}_0}$$
$$\{n\backslash 0\backslash 1\}$$ $$f_{\zeta^{\omega^n}_0}$$
$$\{0,1\backslash 0\backslash 1\}$$ $$f_{\zeta^{\omega^\omega}_0}$$
$$\{0\{1\}1\backslash 0\backslash 1\}$$ $$f_{\zeta^{\omega^{\omega^\omega}}_0}$$
$$\{0\{0,1\}1\backslash 0\backslash 1\}$$ $$f_{\zeta^{\omega^{\omega^{\omega^\omega}}}_0}$$
$$\{0\{0\{....\}1\}1\backslash 0\backslash 1\}$$ $$f_{\zeta^{\omega^{\omega^{\omega^{\omega^{....}}}}}_0}$$
$$\{0\{0\backslash 1\}1\backslash 0\backslash 1\}$$ $$f_{\zeta^{\epsilon_0}_0}$$
$$\{A\backslash 0\backslash 1\}$$ for $$A$$ is an array in the seperator $$f_{\zeta^{\omega^{\lambda}}_0}$$ for $$\lambda$$ is the ordinal corresponding to $$A$$
$$\{0\{0\backslash 0\backslash 1\}1\backslash 0\backslash 1\}$$ $$f_{\zeta^{\zeta_0}_0}$$
$$\{0\{0\{0\backslash 0\backslash 1\}1\backslash 0\backslash 1\}1\backslash 0\backslash 1\}$$ $$f_{\zeta^{\zeta^{\zeta_0}_0}_0}$$
$$\{0\backslash 2\backslash 1\}$$ $$f_{\varepsilon_{\zeta_0 +1}}$$
$$\{A\backslash 2\backslash 1\}$$ $$f_{\varepsilon^{\omega^\lambda}_{\zeta_0 +1}}$$
$$\{0\backslash n+1\backslash 1\}$$ $$f_{\varepsilon_{\zeta_0 +n}}$$
$$\{0\backslash 0,1\backslash 1\}$$ $$f_{\varepsilon_{\zeta_0 +\omega}}$$
$$\{0\backslash 0\{1\}1\backslash 1\}$$ $$f_{\varepsilon_{\zeta_0 +\omega^\omega}}$$
$$\{0\backslash 0\{0\backslash 1\}1\backslash 1\}$$ $$f_{\varepsilon_{\zeta_0 +\varepsilon_0}}$$
$$\{0\backslash 0\{0\backslash 0\backslash 1\}1\backslash 1\}$$ $$f_{\varepsilon_{\zeta_0 2}}$$
$$\{0\backslash 0\{0\backslash 0\backslash 1\}0,1\backslash 1\}$$ $$f_{\varepsilon_{\zeta_0 \omega}}$$
$$\{0\backslash 0\{1\backslash 0\backslash 1\}1\backslash 1\}$$ $$f_{\varepsilon_{\zeta^{\omega}_0}}$$
$$\{0\backslash 0\{0\backslash 1\backslash 1\}1\backslash 1\}$$ $$f_{\varepsilon_{\varepsilon_{\zeta_0 +1}}}$$
$$\{0\backslash 0\{0\backslash 0\{0\backslash 1\backslash 1\}1\backslash 1\}1\backslash 1\}$$ $$f_{\varepsilon_{\varepsilon_{\varepsilon_{\zeta_0}}}}$$
$$\{0\backslash 0\backslash 2\}$$ $$f_{\zeta_1}$$
$$\{0\backslash 0\backslash n+1\}$$ $$f_{\zeta_n}$$
$$\{0\backslash 0\backslash 0,1\}$$ $$f_{\zeta_\omega}$$
$$\{0\backslash 0\backslash 0\{0\backslash 1\}1\}$$ $$f_{\zeta_{\varepsilon_0}}$$
$$\{0\backslash 0\backslash A\}$$ $$f_{\zeta_\lambda}$$
$$\{0\backslash 0\backslash 0\{0\backslash 0\backslash 1\}1\}$$ $$f_{\zeta_{\zeta_0}}$$
$$\{0\backslash 0\backslash 0\{0\backslash 0\backslash 0\{0\backslash 0\backslash 1\}1\}1\}$$ $$f_{\zeta_{\zeta_{\zeta_0}}}$$

For the sake of time and simplicity,I will,from now on,use $$\alpha$$ as a shorthand for $$f_{\alpha}$$.

Up to $$\varphi(\omega,0)$$

$$\{0\backslash 0\backslash 0\backslash 1\}$$ $$\eta_0$$
$$\{1\backslash 0\backslash 0\backslash 1\}$$ $$\eta^{\omega}_0$$
$$\{A\backslash 0\backslash 0\backslash 1\}$$ $$\eta^{\omega^\lambda}_0$$
$$\{0\{0\backslash 0\backslash 0\backslash 1\}1\backslash 0\backslash 0\backslash 1\}$$ $$\eta^{\eta_0}_0$$
$$\{0\{0\{0\backslash 0\backslash 0\backslash 1\}1\backslash 0\backslash 0\backslash 1\}1\backslash 0\backslash 0\backslash 1\}$$ $$\eta^{\eta^{\eta_0}_0}_0$$
$$\{0\backslash 1\backslash 0\backslash 1\}$$ $$\varepsilon_{\eta_0 +1}$$
$$\{0\backslash 0,1\backslash 0\backslash 1\}$$ $$\varepsilon_{\eta_0 +\omega}$$
$$\{0\backslash 0\{0\backslash 0\backslash 0\backslash 1\}1\backslash 0\backslash 1\}$$ $$\varepsilon_{\eta_0 2}$$
$$\{0\backslash A\backslash 0\backslash 1\}$$ $$\varepsilon_{\eta_0 +\lambda}$$
$$\{0\backslash 0\{0\backslash 1\backslash 0\backslash 1\}1\backslash 0\backslash 1\}$$ $$\varepsilon_{\varepsilon_{\eta_0 +1}}$$
$$\{0\backslash 0\backslash 1\backslash 1\}$$ $$\zeta_{\eta_0 +1}$$
$$\{0\backslash 0\backslash 0,1\backslash 1\}$$ $$\zeta_{\eta_0 +\omega}$$
$$\{0\backslash 0\backslash 0\{0\backslash 1\}1\backslash 1\}$$ $$\zeta_{\eta_0 +\varepsilon_0}$$
$$\{0\backslash 0\backslash 0\{0\backslash 0\backslash 0\backslash 1\}1\backslash 1\}$$ $$\zeta_{\eta_0 2}$$
$$\{0\backslash 0\backslash A\backslash 1\}$$ $$\zeta_{\eta_0 + \lambda}$$
$$\{0\backslash 0\backslash 0\{0\backslash 0\backslash 0\backslash 1\}1\backslash 1\}$$ $$\zeta_{\zeta_{\eta_0 +1}}$$
$$\{0\backslash 0\backslash 0\backslash 2\}$$ $$\eta_1$$
$$\{0\backslash 0\backslash 0\backslash n+1\}$$ $$\eta_n$$
$$\{0\backslash 0\backslash 0\backslash 0,1\}$$ $$\eta_\omega$$
$$\{0\backslash 0\backslash 0\backslash 0\{1\}1\}$$ $$\eta_{\omega^\omega}$$
$$\{0\backslash 0\backslash 0\backslash 0\{0\backslash 1\}1\}$$ $$\eta_{\varepsilon_0}$$
$$\{0\backslash 0\backslash 0\backslash A\}$$ $$\eta_\lambda$$
$$\{0\backslash 0\backslash 0\backslash 0\{0\backslash 0\backslash 0\backslash 1\}1\}$$ $$\eta_{\eta_0}$$
$$\{0\backslash 0\backslash 0\backslash 0\{0\backslash 0\backslash 0\backslash 0\{......\}1\}1\}$$ $$\eta_{\eta_{......}}$$
$$\{0\backslash 0\backslash 0\backslash 0\backslash 1\}$$ $$\varphi(4,0)$$
$$\{0\backslash 1\backslash 0\backslash 0\backslash 1\}$$ $$\varepsilon_{\varphi(4,0)+1}$$
$$\{0\backslash 0\backslash 1\backslash 0\backslash 1\}$$ $$\varphi(2,\varphi(4,0)+1)$$
$$\{0\backslash 0\backslash 0\backslash 1\backslash 1\}$$ $$\varphi(3,\varphi(4,0)+1)$$
$$\{0\backslash 0\backslash 0\backslash 0\backslash 2\}$$ $$\varphi(4,1)$$
$$\{0\backslash 0\backslash 0\backslash 0\backslash 0,1\}$$ $$\varphi(4,\omega)$$
$$\{0\backslash 0\backslash 0\backslash 0\backslash A\}$$ $$\varphi(4,\lambda)$$
$$\{0\backslash 0\backslash 0\backslash 0\backslash 0\{0\backslash 0\backslash 0\backslash 0\backslash 1\}1\}$$ $$\varphi(4,\varphi(4,0))$$
$$\{0\backslash 0\backslash 0\backslash 0\backslash 0\backslash 1\}$$ $$\varphi(5,0)$$
$$\{a\backslash b\backslash c.....i\backslash j\}$$ W/ $$n$$ numbers $$\varphi(1,\varphi(2,\varphi(...... \\ \varphi(n-2,\varphi(n-1,j-1)+i).....)+c)+b)^a$$
$$\{0\backslash 0\backslash 0.......0\backslash 1\}$$ W/ $$n$$ zeros $$\varphi(n,0)$$

Limits

The ordinal limits for ABHAN so far go as follows:

• Linear arrays (l-ABHAN) - $$\omega^\omega$$
• Multidimentional arrays (md-ABHAN) - $$\omega^{\omega^\omega}$$
• Hyperdimentional arrays (hd-ABHAN) - $$\varepsilon_0$$

Nested arrays are not yet completely analysed,though it should reach as far as $$\psi(\varepsilon_{\Omega +1})$$,the Bachmann-Howard Ordinal!