The Order Extension Principle (OEP) is the proposition that every partial order on a set can be extended to a linear order on the same set. The OEP is a theorem of ZFC but not of ZF, and it is among the many choice principles that appear in a book such as Axiom of Choice by Herrlich.
Among notions between those of posets and linearly ordered sets is that of directed sets. One can think of an analogue of the OEP but for directed sets: every partial order on a set can be extended to an order that makes the set directed. Is this a theorem of ZF? (The statement was not on the said book.)
from Hot Weekly Questions - Mathematics Stack Exchange
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