# antichain

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## antichain

[′an·tē‚chān]
(mathematics)
A subset of a partially ordered set in which no pair is a comparable pair.

## antichain

(mathematics)
A subset S of a partially ordered set P is an antichain if,

for all x, y in S, x <= y => x = y

I.e. no two different elements are related.

("<=" is written in LaTeX as \subseteq).
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References in periodicals archive ?
A set of pairwise incomparable elements of the root poset is called an antichain.
Those invariants are being a chain, antichain, directed poset, semiorder and satisfying a given inequality.
Let R be the poset product of a k-element chain by a denumerable antichain.
For example, if P is an antichain, then generalized subword order on [P.
is said to be an antichain if A is a subset of [[PHI].
We call S [subset or equal to] P an antichain when x, y [member of] S and x [not equal to] y imply that x and y are incomparable (i.
If A is an antichain of P, then e(P) [greater than or equal to] [[summation].
Let E be a finite set of (unlabelled) connected graphs that forms an antichain with respect to the minor order.
Does there exist an infinite antichain in the Dyck pattern poset?
In particular, in 2007, Panyushev [Pan08] conjectured and in 2011, Amstrong, Stump, and Thomas [AST11] proved that if S is the set of antichains in the root poset of a finite Weyl group, [phi] is the operation variously called the Brouwer-Schrijver map [BS74], the Fon-der- Flaass map [Fon93, CF95], the reverse map [Pan08], Panyushev complementation [AST11], and rowmotion [SW12], and f (A) is the cardinality of the antichain A, then (S, [phi], f) satisfies (2).
4 A (uniform) nonnesting partition for W is an antichain in the root poset of W.
Note that we obtain ordinary factor order by taking P to be an antichain.

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