Monotone boundary representation of a grid order ideal ($B=\{(x,y):0\le x<r_y\},r_0\ge r_1\ge \cdots$)

For an order ideal in a finite rectangular grid with coordinatewise partial order, occupied cells in each row form a left prefix. Their lengths form a nonincreasing sequence $r_y$. Conversely, any nonincreasing sequence of valid row lengths specifies exactly one such order ideal, giving an abstraction function with no redundant per-cell information.

Removing all occupied cells with $x\ge a$ and $y\ge b$ replaces $r_y$ by $\min (r_y,a)$ for $y\ge b$ and leaves earlier rows unchanged. The new lengths remain nonincreasing: affected rows preserve their mutual order, and the first affected row only decreases. For a legal removal, the boundary immediately below it was already at least $a$. A cached sum of row lengths can be updated by subtracting the decreases. This yields constant-time membership and count queries, linear storage, and an update linear in the number of affected rows.