Splitting Properties in Partially Ordered Sets and Set Systems

Ahlswede, Rudolf; Khachatrian, Levon H.

doi:10.1007/978-1-4757-6048-4_4

Cited by 3 publications

(5 citation statements)

References 2 publications

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“…This result is a farfetched generalization of the construction given by Ahlswede and Khachatrian in [2].…”

Section: Corollary 37 [ω]supporting

confidence: 71%

“…Ahlswede and Khachatrian showed ( [2]) that the plain generalization of Theorem 1.1 for infinite posets fails: the finite-subset-lattice [ω] <ω , ⊂ , which is cut-free, contains an infinite antichain which does not split.…”

Section: Theorem 11 ([1 Theorem 31]) Let P Be a Finite Cut-free Pmentioning

confidence: 99%

“…Let us recall that Ahlswede and Khachatrian use number theory in [2] to construct a non-splitting antichain; our proof is purely combinatorial. Besides this result in Section 3 we also investigate possible strengthening of the statements that "A does not split".…”

Section: Theorem 11 ([1 Theorem 31]) Let P Be a Finite Cut-free Pmentioning

confidence: 99%

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How To Split Antichains In Infinite Posets*

Erdős

Soukup

2007

Combinatorica

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A maximal antichain A of poset P splits if and only if there is a setBy [1] every maximal antichain in a finite cut-free poset splits. Although this statement for infinite posets fails (see [2])) we prove here that if a maximal antichain in a cut-free poset "resembles" to a finite set then it splits. We also show that a version of this theorem is just equivalent to Axiom of Choice.We also investigate possible strengthening of the statements that "A does not split" and we could find a maximal strengthening.An antichain in P is a set of pairwise incomparable elements. If A is a maximal antichain in P then clearly P = A ↑ ∪ A ↓ . We say that A splits if

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“…This result is a farfetched generalization of the construction given by Ahlswede and Khachatrian in [2].…”

Section: Corollary 37 [ω]supporting

confidence: 71%

Section: Theorem 11 ([1 Theorem 31]) Let P Be a Finite Cut-free Pmentioning

confidence: 99%

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How To Split Antichains In Infinite Posets*

Erdős

Soukup

2007

Combinatorica

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“…In particular, if B ∩ ↓A = ∅, it can be split into two subsets (A and B) so that every element of L \ C is above A or below B. This fact has a direct connection to the splitting property of posets (see [1,2,6,9,10,11,13]). Indeed, in [14] it is proved that finite maximal antichains in the poset arising from the category C of relational structures with one relation contain no elements of ↓wld( C) \ wld( C).…”

Section: Sparse Incomparability and Antichainsmentioning

confidence: 99%

Dualities and Dual Pairs in Heyting Algebras

Foniok

Nešetřil

Pultr

et al. 2010

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“…The splitting property of maximal antichains in general posets has been extensively studied, see [1,2,4,5,6,8,9,10]. Here we are concerned with the splitting property of maximal antichains in the homomorphism order of finite relational structures.…”

Section: Introductionmentioning

confidence: 99%