# I have a question about infinite partially ordered sets.

If I have an infinite partially ordered set $P$ where every chain has finite order and I take some maximal element $x_1$ (which must exists because our chains are finite) and then I take a maximal element $x_2\in P-{x_1}$ is it true that $\{x_1,x_2\}$ is an anti chain?

I don't think it is but I am not quite sure to find a counterexample.

What I think is true, but not quite sure how to prove it. $\downarrow$

If I take some element $x_1\in P$ and then consider its maximal chain $T_1$ and consider the maximal element $t_1\in T_1$ and then take some $x_2\in P-T_1$ and consider its maximal chain $T_2$ with maximal element $t_2$ and continue this, then $\{t_1,t_2,t_3...\}$ is an anti chain.

Is any of this true?

• What if $x_2$ is in the same chain as $x_1$? – Git Gud Oct 16 '14 at 9:10
• OK what about the one I thought might be true? – H_B Oct 16 '14 at 9:13
• the chains have finite elements, there are infinite chains because every singleton is a chain. – H_B Oct 16 '14 at 9:20
• what I am trying to do is suppose there are only chains with finite amount of elements, and prove that there must be an infinite anti-chain. – H_B Oct 16 '14 at 9:21
• Now I get it. And I misread your last paragraph at first. I think your idea works. – Git Gud Oct 16 '14 at 9:24

If $x_1$ is a maximal element of $P$ and $x_2\in P-x_1$, is $\{x_1,x_2\}$ an antichain? No, this fails when $P$ is a $2$-element chain.
"Take some element $x_1$ and consider its maximal chain $T_1$." What do you mean by "its" maximal chain? You mean, a maximal chain $T_1$ containing $x_1$, picked at random from the many maximal chains containing $x_1$? (Fine, but since the element $x_1$ isn't mentioned again, why bother with $x_1$? Why not just choose a maximal chain $T_1$?) Then $T_1$ will have a maximal element $t_1$, since all chains in $P$ are finite; and then $t_1$ is a maximal element of $P$, since $T_1$ is a maximal chain.
Are you going to get an infinite antichain this way? Not necessarily. What if $P$ happens to have a greatest element? Whatever you take for $x_1$ and $T_1$, you're going to end up with that top element of $P$ as your $t_1$. Needless to say, that $t_1$ is not going to be part of an infinite antichain. Sorry!