Critical pair (order theory)
In order theory, a discipline within mathematics, a critical pair is a pair of elements in a partially ordered set that are incomparable but that could be made comparable without requiring any other changes to the partial order. Formally, let  be a partially ordered set. Then a critical pair is an ordered pair  of elements of  with the following three properties:
 and  are incomparable in ,
for every  in , if  then , and
for every  in , if  then . If  is a critical pair, then the binary relation obtained from  by adding the single relationship  is also a partial order. The properties required of critical pairs ensure that, when the relationship  is added, the addition does not cause any violations of the transitive property. A set  of linear extensions of  is said to reverse a critical pair  in  if there exists a linear extension in  for which  occurs earlier than . This property may be used to characterize realizers of finite partial orders: A nonempty set  of linear extensions is a realizer if and only if it reverses every critical pair. References
. Order theory