Someone please can help me out ? Please. I am trying to use the definition of product ordinal and exponent ordinal.
Product ordinal: $\beta \cdot \alpha=\sup \{\beta \cdot \delta: \delta<\alpha\}$
Exponent ordinal: $\beta^\alpha=\sup \left\{\beta^\delta: \delta<\alpha\right\}$
can I define this one, in this way?
$\omega \cdot \omega^{\omega}=\sup\left\{\omega \cdot \omega^\alpha: \omega^\alpha<\omega^{\omega}\right\}.$
Is that possible?
$\omega \cdot \omega^{\omega} = \omega^{1} \cdot \omega^{\omega} = \omega^{1+\omega} = \omega ^ {\omega}$
Note that for ordinals, $\alpha, \beta, \gamma$, it is true that $\alpha^{\beta} \cdot \alpha^{\gamma} = \alpha^{\beta + \gamma}$.