It seems to me that the dual of Zorn's Lemma should be true: if $S$ is a non-empty partially ordered set and every chain of $S$ has a lower bound in $S$, then $S$ has at least one minimal element.

Zorn's Lemma comes up quite often in Commutative Algebra, but I don't recall ever seeing a dual of it or any applications of a dual. Are there any interesting applications of the dual, assuming it is true?

P.S. I am not a logician.

  • 16
    $\begingroup$ The principle is equivalent to Zorn's lemma, if you just turn the order upside-down. $\endgroup$ – Joel David Hamkins Sep 15 '10 at 1:23
  • $\begingroup$ This is too easy to be a homework problem. I vote to close. $\endgroup$ – Bill Johnson Sep 15 '10 at 1:46
  • $\begingroup$ Hannay, in my opinion this sort of question belongs on math.stackexchange.com and not here. $\endgroup$ – Asaf Karagila Sep 15 '10 at 1:47
  • $\begingroup$ Maybe a more interesting version of the question is: If we take the axiom of choice to mean "Every epi has a right inverse," what can be said of the "axiom", "Every monic has a left inverse"? Of course I haven't thought about this at all, but my guess is that this would fail in a lot of topoi? $\endgroup$ – Dylan Wilson Sep 15 '10 at 2:22
  • 1
    $\begingroup$ @Dylan Wilson: I agree that that is a more interesting "dual to axiom of choice". But I disagree that it is a "version of the question" that OP asked. $\endgroup$ – Theo Johnson-Freyd Sep 15 '10 at 5:45

If your poset is $\langle S,<\rangle$ and it has the property that every chain has a lower bound, define the order $R$ on $S$ which is: $aRb \iff b< a$.

It is easy to see that every chain has an upper bound, satisfying Zorn's lemma and that the maximal element in $R$ is the minimal element in the original order.


Not the answer you're looking for? Browse other questions tagged or ask your own question.