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:
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.