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1 Chains and Antichains Form
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People also ask
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What is the difference between a chain and a poset?
A set with a partial ordering is called a partially ordered set or a poset. A poset with every pair of distinct elements comparable is called a totally ordered set. A total ordering is also called a linear ordering, and a totally ordered set is also called a chain. 7.4: Partial and Total Ordering - Mathematics LibreTexts libretexts.org https://math.libretexts.org › Bookshelves › 07:_Relations libretexts.org https://math.libretexts.org › Bookshelves › 07:_Relations
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How to find the largest antichain?
Because of theorem 1.3. 4 we know that among all anti-chains of the form [nk] the largest are the "middle'' ones, namely [n⌊n/2⌋] and [n⌈n/2⌉] (which are the same if n is even). Remarkably, these are the largest of all anti-chains, that is, strictly larger than every other anti-chain. 1.7 Sperner's Theorem whitman.edu https://.whitman.edu › book › section01.07.html whitman.edu https://.whitman.edu › book › section01.07.html
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How to find chain and antichain?
A subset C ⊆ P is a chain if (C,<) is linearly ordered. A subset A ⊆ P is an antichain if no two distinct elements of A are comparable under <. It follows from Ramsey's theorem for pairs that every infinite partial ordering has an infinite chain or an infinite antichain.
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What is the number of antichains in a poset?
Every antichain contains at most one vertex from each of these chains, so the number of antichains is at most (c1+1)(c2+1)⋯(ca+1)≤(n/a+1)a, where n is the number of elements of the poset.
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What is an example of a maximal antichain?
Find the number of maximal antichains in the set {1,2,3,4,5,6,7,...,120} where the order is divisibility relation. For example, {6,7,15} is an antichain but not a maximal antichain, and {1} is a maximal antichain, and {p:p is a prime less than 120} is a maximal antichain.
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What is an example of a maximal antichain?
Find the number of maximal antichains in the set {1,2,3,4,5,6,7,...,120} where the order is divisibility relation. For example, {6,7,15} is an antichain but not a maximal antichain, and {1} is a maximal antichain, and {p:p is a prime less than 120} is a maximal antichain. Number of maximal antichains in the set $\{1,2,3,4,5,6,...,120\}$ where ... stackexchange.com https://math.stackexchange.com › questions › number-of... stackexchange.com https://math.stackexchange.com › questions › number-of...
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What is a chain in set theory?
A chain in is a set of pairwise comparable elements (i.e., a totally ordered subset). The partial order length of is the maximum cardinal number of a chain in. . For a partial order, the size of the longest chain is called the partial order length. Chain -- from Wolfram MathWorld wolfram.com https://mathworld.wolfram.com › Chain wolfram.com https://mathworld.wolfram.com › Chain
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What is a chain and an antichain?
A chain in is a subset in which each pair of elements is comparable; that is, is totally ordered. An antichain in is a subset of. in which each pair of different elements is incomparable; that is, there is no order relation between any two different elements in.
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