Why Zorn's Lemma "fails" on nested boxes problem?

Question

Backgound

My problem comes from the nested boxes problem. Consider a list of boxes. Each box has a length, width, and height. Since the boxes can be rotated those terms are interchangeable. The boxes have rectangular surfaces and can be nested inside each other. A box can nest inside another box if all its dimensions are less than the corresponding dimensions of the other. You may only nest a box such that the corresponding surfaces are parallel to each other. A box may not be nested along the diagonal. You cannot also put two or more boxes side by side inside another box.

Therefore, a box can be represented by a sorted (it's not hard to show that sorting the dimensions in ascending order won't affect the result of whether a box can be nested inside another) tuple of dimensions. The set of boxes and the relation of being able to nested in another actually form a partially ordered set. And here comes my problem:

Formalization

Let's use an example. Let $S=\{(1,1,1),\,(3,4,4),\,(2,5,5)\}$. We define relation $\le$ to be $$ (x_1,y_1,z_1)\le(x_2,y_2,z_2) \text{ if and only if } x_1\le x_2,\,y_1\le y_2,\,\text{and }z_1\le z_2 $$ We see that $(S,\le)$ is a partially ordered set.

Now, all the totally ordered subset of $S$ are: $$ S_1=\{(1,1,1),\,(3,4,4)\}\quad S_2=\{(1,1,1),(2,5,5)\} $$ where $(3,4,4)$ is the upper bound of $S_1$, $(2,5,5)$ is the upper bound of $S_2$.

However, Zorn's Lemma says:

Suppose a partially ordered set $P$ has the property that every chain (i.e. totally ordered subset) has an upper bound in $P$. Then the set $P$ contains at least one maximal element.

Therefore, $S$ must have a maximal element. But it's obvious that $S$ doesn't have one.

Did I miss something here? Thanks.

Answer 1

Both $\langle 3,4,4\rangle$ and $\langle 2,5,5\rangle$ are maximal elements of $S$: a maximal element is simply one that is not strictly less than any other element. Zorn’s lemma does not guarantee that $S$ has a maximum element, which I suspect is what you have in mind.

(By the way, $S$ has three other non-empty chains: they just aren’t maximal chains. Specifically, the other non-empty chains are the three singleton subsets.)

Answer 2

I suspect you are confused as to the definition of maximal element. A maximal element of a partially ordered set $(S,\leq)$ is an element $s\in S$ such that if $t\in S$ and $s\leq t$, then $s=t$. It should be clear that the maximal elements of $S$ in your example are $(3,4,4)$ and $(2,5,5)$.