My Math Forum Extrapolating a partial ordering

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 January 21st, 2018, 09:02 AM #1 Senior Member   Joined: Mar 2012 From: Belgium Posts: 654 Thanks: 11 Extrapolating a partial ordering Suppose we have a set $\displaystyle S$ and an order $\displaystyle \leq$ on most of the elements $\displaystyle s_1,s_2 \in S$. The problem is that the order is not defined for all pairs in $\displaystyle S$. Now how would I go and find an order $\displaystyle \preceq$ on S such that $\displaystyle s1 \preceq s2$ if $\displaystyle s1 \leq s2$ and such that all elements of $\displaystyle S$ are comparable for this order. What happens with contradictions in the original order ? Last edited by gelatine1; January 21st, 2018 at 09:05 AM.
 January 21st, 2018, 09:53 AM #2 Senior Member   Joined: Oct 2009 Posts: 608 Thanks: 186 Zorn's lemma.

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