Chain-complete poset

A poset $(P,\le )$ is chain-complete if every chain $C\subseteq P$ (Chain in a poset) has a least upper bound $\sup C$ in $P$ (Upper bound, least upper bound, and strict upper bound).

The empty set is a chain, so a chain-complete poset has an element $\mathrm{\perp }:=\sup \mathrm{\varnothing },$ and $\mathrm{\perp }$ is the least element of $P$: every $x\in P$ is an upper bound of $\mathrm{\varnothing }$, so $\mathrm{\perp }\le x$ by leastness. In particular a chain-complete poset is nonempty.

A map $f:P\to P$ is progressive (also inflationary, or increasing in Bourbaki's sense) if $x\le f(x)\text{for every }x\in P\mathrm{.}$

• Progressive is not monotone. A progressive map is required to move each point weakly upward; it is not required to preserve the order, and Bourbaki–Witt fixed point theorem assumes no monotonicity whatsoever. This is what makes the theorem strong enough to drive Zorn's lemma, where the map is built from an arbitrary choice function and has no reason to be monotone.
• On the empty-chain convention. Some authors let "chain" mean nonempty chain, and then state Bourbaki–Witt for a nonempty chain-complete poset. The two conventions give the same theorem. Given $P\mathrm{\ne }\mathrm{\varnothing }$ chain-complete in the nonempty-chain sense and $f$ progressive, pick any $p\in P$ and pass to $P_{\ge p}=\{x\in P:p\le x\}$: it contains $p$, it is closed under $f$ because $f$ is progressive, the supremum of a nonempty chain in $P_{\ge p}$ again lies in $P_{\ge p}$, and $\sup \mathrm{\varnothing }=p$ there. So $P_{\ge p}$ is chain-complete in the sense used here. Including the empty chain simply packages that reduction into the definition, and it is Wikipedia's convention for a pointed chain-complete order.
• Chain-completeness is strictly weaker than requiring least upper bounds for all subsets (which would make $P$ a complete lattice). The power set of any set is a complete lattice, hence chain-complete (The power set is chain-complete, with union as supremum ↗); the posets Zorn's lemma is applied to usually are not.
• The hypothesis cannot be dropped from Bourbaki–Witt fixed point theorem: a progressive map on a poset that is not chain-complete may have no fixed point (A progressive map with no fixed point, on a poset that is not chain-complete ↗).