User blog comment:Ikosarakt1/Fast-growing hierarchy/@comment-5529393-20130627000838/@comment-5529393-20130628034551

Actually, \((\Omega^2 + \Omega) * \Omega = \Omega^3\). (To see this, observe that the left hand side is the limit of \((\Omega^2 + \Omega) * \alpha\) for countable \(\alpha\), and that expression always has value less than \(\Omega^3\).)  So the rules won't work whether you use # or \(\alpha\).

I would suggest expressing the ordinal as \(\alpha + \Omega^{\beta} * \gamma\), and define rules based on the cofinality of \(\beta\) and \(\gamma\). (The cofinality of 0 is 0, the cofinality of a successor ordinal is 1, and otherwise the cofinality will be \(\omega\) or \(\Omega\) depending.)