Partial Orders on finite sets to which Zorn's lemma does not apply Zorn's lemma is stated in the text I'm reading as

If $(A, \leq)$ is a poset such that every chain of elements in $A$ has an upper bound in $A$, then $A$ has at least one maximal element.

However, I can't think of any partial ordering (defined as a relation which is reflexive, anti-symmetric, and transitive) on which the condition

If $(A, \leq)$ is a poset such that every chain of elements in $A$ has an upper bound in $A$

would not hold if $A$ is a finite set. (I understand how an infinite set like $[0, 1)$ would fail to satisfy this requirement). My argument is that if $C$ is a chain in the poset $(A, \leq)$ then there will be an end to the chain which will be an element in the chain, and hence every set will satisfy this.
Then can we say that if $A$ is a finite set then any poset $(A, \leq)$ has at least one maximal element by Zorn's lemma? If not, is there a counter-example to help me understand?
 A: Well, there is a counterexample, but it's not a very interesting one: if $A=\emptyset$, then the empty chain has no upper bound.  However, any nonempty finite poset satisfies the conditions of Zorn's lemma.  Indeed, any finite nonempty chain has a greatest element (this is easy to prove by induction on the size of the chain), which is then an upper bound.  (And as long as $A$ is nonempty, the empty chain also has an upper bound, namely any element of $A$.)
As others have mentioned, Zorn's lemma is kind of overkill for finite posets, because you can prove they have maximal elements without the axiom of choice.  However, I think they are actually an instructive example for understanding the proof of Zorn's lemma in general.  There is an obvious way to find a maximal element of a (nonempty) finite poset: start with one chosen element, and then compare it with all the elements of the poset one by one, changing your chosen element whenever you find another element that is greater than it.  This argument is in fact exactly how you prove Zorn's lemma in the general case as well, except that you need to choose a well-ordering of the poset and use it to go through the elements one-by-one by transfinite induction, and at limit steps you need to use the hypothesis that chains have upper bounds.
A: I actually wrote , many years ago , an example that may satisfy your curiosity:
https://coldoc.sns.it/UUID/EDB/02M/?lang=en_US
Indeed in this example there is a partial order that cannot satisfy Zorn's Lemma hypotheses, but it is not obvious at first sight.
