Extremal element and its cut (Bourbaki–Witt)

Let $(P, \le)$ be a chain-complete poset, $f : P \to P$ progressive, and let $M$ be the smallest $f$-admissible subset of $P$ (A smallest admissible set exists).

An element $x \in M$ is extremal if $\text{for every } y \in M \text{ with } y < x, \quad f(y) \le x.$

For $x \in M$, the cut at $x$ is the subset $M_x := \{\, y \in M \ : \ y \le x \ \text{ or } \ f(x) \le y \,\} \subseteq M.$

• Read extremality as: $x$ cannot be jumped over from below. If $y$ sits strictly below $x$ inside $M$, then applying $f$ to $y$ does not carry it past $x$. Nothing in the definition says such $y$ exist; $\bot$ is extremal vacuously.
• The cut $M_x$ is the set of elements of $M$ that are comparable to $x$ in the strong sense of lying at or below $x$, or at or above $f(x)$. It deliberately omits anything strictly between $x$ and $f(x)$. The whole Bourbaki–Witt argument consists of showing that for extremal $x$ the cut is everything (Everything in $M$ is comparable to an extremal element) and that every element is extremal (Every element of $M$ is extremal), and those two facts together say precisely that $M$ is totally ordered (The smallest admissible set is a chain).
• Both notions are relative to $M$ and to $f$, not to $P$. They are scaffolding for one proof and are not used after Bourbaki–Witt fixed point theorem.