# 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 ?
In Beard's words, "Planned production without rationalized retailing is an impossibility." (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. The support of a root [beta] [member of] [[PHI].sup.+] is the set of all [alpha] [member of] S with [alpha] [less than or equal to] [beta].
For this purpose, the concepts of antichain and blocker are introduced.
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. More formally, R := (E, [rho]) where E := {1,..., k} x [??] and [rho] := {((i, n), (j, m)) [member of] E such that i [less than or equal to] j}.
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.
It is obtained by (e.g., antichain) powerset completion and reduction (viz., sets of basic comportments denoting the same object in ??
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.sup.*] is equivalent to ordinary subword order.
The pharmacy laws were decidedly antichain. And any attempts to promote chain drug pharmacy, such as Osco's bold step in the early 1970s of posting the prices of the 100 best-selling prescription drugs, were viewed as misguided - or unprofessional.

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