Definition ($lambda$-nice names). Generalise “nice names” to $\lambda$-nice names:

Let

$$\mathrm{OL}=\{A_{\alpha }:\alpha <\lambda \}$$

be a family of $\lambda$ many maximal $\mathbb{P}$-antichains:

$${\tau }_{\mathrm{OL}}:=\{(\check{\alpha },p):p\in A_{\alpha }\}$$

for $\alpha \in \lambda$.

These are names for subsets of $\lambda$.