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The term "critical pair" has two distinct meanings in mathematics: in mathematical logic and in order theory.
## Usage in mathematical logic

A critical pair arises in term rewriting systems where rewrite rules overlap to yield two different terms. _{1} yields the term g(x,x), while applying ρ_{2} yields the term f(x,x). The critical pair is then (g(x,x), f(x,x)).## Usage in order theory

A critical pair in a partially ordered set P = (X, ≤) is, loosely speaking, a pair of elements that are very nearly, but not quite, comparable. Formally, it is an ordered pair (x, y) of elements of X that are incomparable in P and such that for all z in X, if z < x then z < y and if y < z then x < z. Among the reasons that critical pairs are of interest is the fact that a family F of linear extensions of any poset P is a realizer of P if and only if for every critical pair (u, v) of P there is some linear extension L in F for which u <_{L} v.
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## References

For example, in the term rewriting system with rules

- $rho\_1\; :\; f(g(x,y),\; z)\; rightarrow\; g(x,z)$

- $rho\_2\; :\; g(x,y)\; rightarrow\; x$,

When one side of the critical pair reduces to the other, we say that the critical pair is convergent. Where one side of the critical pair is identical to the other, we say that the critical pair is trivial.

Note that if the term rewriting system is not confluent, the critical pair may not converge, so critical pairs are potential sources where confluence will fail. In fact, we have the critical pair lemma, which states that a term rewriting system is weakly confluent if all critical pairs are convergent. Weak confluence implies convergent critical pairs clearly as if any critical pair, say (a, b) arises, then a and b have common reduct and thus the critical pair is convergent.

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

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Last updated on Friday July 18, 2008 at 12:28:24 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Friday July 18, 2008 at 12:28:24 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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