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June 8th, 2010, 08:42 PM   #1
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A poset exercise

I'm looking at a (rather old) lattice theory book, and in the first section there's this exercise:

Let ? be a partial order on a set A. Show that there is a total order ?* such that a?b imples a?*b.
(The author defines a partial order as a transitive, reflexive, anti-symmebtric relation, and a total order with the additional property that a?b or b?a.)

The hint says to use Zorn's Lemma. While I can do this with liberal application of choice,this isn't particularly clean, and I'm judging by the hint (and the fact that choice has not been mentioned yet int he book) that there's a way to use Zorn's Lemma directly. I'd also guess this solution is much prettier.

Any ideas on how to go about using Zorn's Lemma directly?
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