User blog comment:Wythagoras/Friendship function/@comment-1605058-20140717170937/@comment-1605058-20140717194603

To bracket \((f_{k_1},f_{k_2}, ... f_{k_t})\) we associate an ordinal \(\omega^{k_1}+\omega^{k_2}+...+\omega^{k_t}\). Because we write brackets so that \(k_1>k_2>...\) we have a well formed ordinal. Nicely, it's a bijection between set of brackets and \(\omega^\omega\). Let \((\alpha)\) denote bracket with ordinal \(\alpha\) associated.

Now to set of brackets \((\alpha_1)(\alpha_2)...(\alpha_k)\) we associate ordinal \(\omega^{\alpha_1}+\omega^{\alpha_2}+...+\omega^{\alpha_k}\). This now gives injection from friendships, forming subset of sets of brackets, to \(\omega^{\omega^\omega}\). I'm fairly sure that if one friendship is minor of another, then associated ordinal is smaller. Converse doesn't hold (there are pairs of friendships with neither being minor of another).

Thus partial order of friendship minorship can be extended to well order of order type w^w^w. This isn't full argument by itself, but it gives strong argument to believe that this function is quite limited, say by w^w^w^w.