DISCLAIMER I am sure this is a duplicate yet I coudn’t find an answer I was looking for so I’m probably going to ask and then eventually flag it as a duplicate.
Is every instance of ordinal multiplication commutative?
More specifically: if we had $\alpha^2\cdot\alpha^\beta$, where both $\alpha$ and $\beta$ are transfinite ordinals, does this give $\alpha^{\beta+2}$ or $\alpha^{2+\beta}$ which is just $\alpha^\beta$.
Ordinal multiplication, like ordinal addition, is noncommutative.
$$2\omega=\omega\lt\omega+\omega=\omega2$$
$$\alpha^{\beta+\gamma}=\alpha^\beta\alpha^\gamma$$
$$\omega^2\omega^\omega=\omega^{2+\omega}=\omega^\omega\lt\omega^{\omega+2}=\omega^\omega\omega^2$$