On Sums of Computable Ordinals

It was proved, that for any computable ordinal $\alpha$ there are notation $a \in O$ and partially computable function with propery: for two notations $b$ and $c$ from set $\{ t \in O \mid t <_{O} a \}$ for ordinals $\beta$ and $\gamma$, $\beta + \gamma < \alpha$, it can find notation for $\beta + \gamma$ from this set. And we show, that not all notations for ordinals $\alpha \geqslant \omega^{2}$ has this property.