Hausdorff maximal principle and Zermelo's fixed point theorem

Question

Let's recall the Hausdorff maximal principle (HPM) and the Zemelo fixed point theorem (ZFPT):

HPM: Every partially ordered set has a maximal chain.

ZFPT: Let $P$ be a partial order set in which every chain has a supremum and $f:P\to P$ such that $x\le f(x)$ for all $x\in P$. Then $f$ has a fixed point.

It is known from Dunford, Schwartz (Linear operators) that ZFPT implies HP, and my question if the converse holds too ?

Hint: let $C$ be a maximal chain such that $\alpha=\sup(C)$, the we have $\alpha\le f(\alpha)$. Hence, $C\cup\{f(\alpha)\}$ is a chain, so by maximality of the chain $C$, we should have $f(\alpha)\in C$ and $f(\alpha)\ge \sup(C)$, which implies $f(\alpha)=\alpha$. Is this correct?

Answer 1

Of course. And yes, your proof is correct.

From the Hausdorff Maximality Principle, we get a maximal chain; from the assumption on upper bounds, this chain has a supremum $x$, which now has to be a maximal element of the partial order. But this means that $f(x)=x$ by maximality.