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January 21st, 2018, 08:02 AM   #1
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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 08:05 AM.
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January 21st, 2018, 08:53 AM   #2
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Zorn's lemma.
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