Is this kind of addition of ordinals incremental?

Question

I just want to ask about if in ZFC the following form of sum operation on ordinals fulfills the below mentioned properties.

Define: $\alpha +^+ \beta = max (\alpha + \beta, \beta + \alpha)$

where: $max(\alpha,\beta)=\gamma \equiv_{df} \\(\alpha > \beta \land \gamma=\alpha) \lor (\beta > \alpha \land \gamma=\beta) \lor (\alpha = \beta \land \gamma = \alpha)$

, and $``+"$ is the usual ordinal adddition.

Does it follow that:

$\gamma > \delta \to \gamma +^+ \alpha > \delta+^+\alpha$

$\gamma \neq \delta \to \gamma +^+ \alpha \neq \delta +^+ \alpha$

Answer 1

We do not get strict inequalities: Consider $\gamma:=\omega^2+1>\delta=\omega^2$ and $\alpha=\omega$. We have $$\gamma+\alpha=\delta+\alpha>\alpha+\gamma>\alpha+\delta.$$