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 $\text{for every }y\in M\text{ with }y<x,f(y)\le x\mathrm{.}$

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

• 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; $\mathrm{\perp }$ 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.