# 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 ?
89) And two years later, when litigants challenged a similar but more extensive antichain law before the Supreme Court, they attached Beard's argument to their brief.
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.
While the legislation never materialized, the antichain sentiment hung on for a long time with Boston residents, extending to other retail trade classes, from the discount arena to department stores.
Recall that a collection of sets is called an antichain if no set in the collection contains another.
Until now its competitors were primarily the nation's 700 independent drug stores, a trade class enhanced by antiquated antichain laws designed to inhibit chain competition by limiting both the number of drug stores any group could own and the proximity of a new drug store to an existing one.
By its minimality, the set B must form an antichain under [less than or equal to], but since infinite antichains are know to exist in the containment partial order, B need not be finite.
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].
An antichain is a subset A of a partially ordered set such that any two elements in A are incomparable.
Executive director for the past 10 years, Wickham, in the minds of many throughout the state, holds antichain views and has a special disdain for Rite Aid.

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