Critical pair (order theory)

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 changing the order relationships of any other pairs of elements.

Formally, let $P = (S, \leq)$ 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 $z < y$ , and

for 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 order relation $x \leq y$ is also a partially ordered set. The required properties of a critical pair ensure that, when the relation $x \leq y$ 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 every critical pair" if, for every critical pair $(x, y)$ of $P$ , 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.

References

Trotter, W. T. (1992), Combinatorics and partially ordered sets: Dimension theory, Johns Hopkins Series in Mathematical Sciences, Baltimore: Johns Hopkins Univ. Press .

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Order theory

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