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• Critical_pair_(order_theory) abstract "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 P = (S, ≤) be a partially ordered set. Then a critical pair is an ordered pair (x, y) of elements of S with the following three properties:x and y are incomparable in P,for every z in S, if z < x then z < y, andfor every z in S, if y < z then x < z.If (x, y) is a critical pair, then the binary relation obtained from P by adding the single relationship x ≤ y is also a partial order. The properties required of critical pairs ensure that, when the relationship x ≤ y is added, the addition does not cause any violations of the transitive property.A set R of linear extensions of P is said to reverse a critical pair (x, y) in P if there exists a linear extension in R for which y occurs earlier than x. This property may be used to characterize realizers of partial orders: A nonempty set R of linear extensions is a realizer if and only if it reverses every critical pair.".
• Critical_pair_(order_theory) thumbnail Critical_pair_(order_theory).svg?width=300.
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• Critical_pair_(order_theory) wikiPageRevisionID "589025233".
• Critical_pair_(order_theory) hasPhotoCollection Critical_pair_(order_theory).
• Critical_pair_(order_theory) subject Category:Order_theory.
• Critical_pair_(order_theory) comment "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 P = (S, ≤) be a partially ordered set.".
• Critical_pair_(order_theory) label "Critical pair (order theory)".
• Critical_pair_(order_theory) sameAs m.0gy0763.
• Critical_pair_(order_theory) sameAs Q5186745.
• Critical_pair_(order_theory) sameAs Q5186745.
• Critical_pair_(order_theory) wasDerivedFrom Critical_pair_(order_theory)?oldid=589025233.
• Critical_pair_(order_theory) depiction Critical_pair_(order_theory).svg.
• Critical_pair_(order_theory) isPrimaryTopicOf Critical_pair_(order_theory).