Sufficient conditions for the existence of $\mathbf0'$-limitwise monotonic functions for computable $\eta$-like linear orders

We find new sufficient conditions for the existence of a $\mathbf0'$-limitwise monotonic function defining the order for a computable $\eta$-like linear order $\mathscr L$, i.e., of a function $G$ such that $\mathscr L\cong\sum_{q\in\mathbb Q}G(q)$. Namely, we define the notions of left local maximal block and right local maximal block and prove that if the sizes of these blocks in a computable $\eta$-like linear order $\mathscr L$ are bounded then there is a $\mathbf0'$-limitwise monotonic function $G$ with $\mathscr L\cong\sum_{q\in\mathbb Q}G(q)$.