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When confronted with finding an object that is maximal with regard to some ordering relation, most of us have the reflex to use Zorn's Lemma.

I am interested in instances of proving the existence of maximal objects, where Zorn's Lemma is explicitly of no use. By that I mean that you can construct chains of objects similar to what you are looking at, and these chains have no upper bound -- but you can prove with other means that maximal objects still do exist. The only example that comes to mind is this, and I am interested in seeing other examples.

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    $\begingroup$ Is this not a duplicate of the linked question? $\endgroup$
    – Emil Jeřábek
    Commented Oct 19, 2017 at 13:52
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    $\begingroup$ It was admonished that the title was unclear, or that the question was not clearly worded. I don't seem able to edit it, so I tried wording it in a better way $\endgroup$
    – Dominic van der Zypen
    Commented Oct 19, 2017 at 14:02
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    $\begingroup$ Basically, you are asking for partially ordered sets in which there are both maximal chains with and without a maximum. In many cases, your proof will simply depend on the fact that the specific maximal chain containing your maximal element is better behaved, and for that chain, ZL must of course hold. $\endgroup$
    – Michael Greinecker
    Commented Oct 19, 2017 at 14:49

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I do not know if this is an example, but it seems related:

Michael Roddy (and others), if I remember correctly, have proven results using the fixed point property (FPP) for posets that seem to enable them to deduce the existence of maximal elements. It may be, though, that those results are themselves based on results that use Zorn's Lemma (namely, Anne Davis's result that a lattice has FPP only if it is complete, a converse to Tarski's Fixpoint Lemma).

See, for example, Corollary 2.4 of Michael S. Roddy, "Fixed Points and Products: Width 3," Order 19 (2002), 319-326.

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