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I was reading Stong's Cobordism Theory, the following lemma to be precise.enter image description here In the proof, he gives a total order on the set of non-dyadic partitions with the given partial order After some research, I came to know that we can always give a total order to any set courtesy this. But could not find anything about the existence of a Total order, Compatible with a given Partial order.

As I am not really an expert on Discrete Mathematics.

Any hints, suggestions will be really helpful.

regards

bof
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Devendra Singh Rana
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1 Answers1

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Let $P$ be a partially ordered set; we want to find a total order $\lt$ which extends the given partial order. This is trivial if $P$ is finite; suppose $P$ is infinite.

Let $\mathcal F$ be the set of all finite subsets of $P$. For each set $F\in\mathcal F$ choose a total order $\lt_F$ of $F$ which is compatible with the given partial order.

Let $\mathcal U$ be an ultrafilter on $\mathcal F$ with the property that $\{F\in\mathcal F:x\in F\}\in\mathcal U$ for each point $x\in P$.

Finally, for $x,y\in P$, define $$x\lt y\iff\{F\in\mathcal F:x\lt_F y\}\in\mathcal U.$$ It is easy to see that $\lt$ is a total order extending the given partial order.

bof
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