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On maximal antichains containing no set and its complement

1981
*
Discrete Mathematics
*

J'e:Tolci Griggs

doi:10.1016/0012-365x(81)90267-3
fatcat:eimeprubnfaxxfhakcttfhfxdy
*and*the referee for some fruitful wggestions.*On*rnaxi~tnal*antichains*con aining*no**set*1 ml*its*cmnpkment ... of S if ;*And*only if X*contains*1.:~ ;tctly*one*vector from each pair. ... or*its**complement*:(i) if n is odd! ...##
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A splitting property of maximal antichains

1995
*
Combinatorica
*

In any dense poset P (e.g. in the Boolean lattice) every

doi:10.1007/bf01192520
fatcat:mr547nzlrja7xgpdkxqh7omo4m
*maximal**antichain*S may be partitioned into disjoint subsets S 1*and*S 2 , such that the union of the downset of S 1 with the upset of S 2 yields ... To find a similar splitting of*maximal**antichains*in posets is NP-hard in general. ...*It*is easy to find a poset P*and*a*maximal**antichain*S in P such that S has*no*splitting. ...##
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Chains, Antichains, and Complements in Infinite Partition Lattices
[article]

2017
*
arXiv
*
pre-print

We consider the partition lattice Π_κ

arXiv:1501.05284v4
fatcat:rqkjpehqxra63efgpujgsre57i
*on*any*set*of transfinite cardinality κ*and*properties of Π_κ whose analogues do not hold for finite cardinalities. ... Moreover, we give a direct formula for the number of*complements*to a given partition; (VI) Under the Generalized Continuum Hypothesis, the cardinalities of*maximal*chains,*maximal**antichains*,*and*numbers ... The authors thank Benoît Kloeckner for his many comments*on*a preliminary version of this article,*and*thank the anonymous referees for valuable suggestions. ...##
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Note on maximal split-stable subgraphs

2007
*
Discrete Applied Mathematics
*

In this note, we address the corresponding optimization problem: identify a

doi:10.1016/j.dam.2007.04.009
fatcat:gxgqqq5tkzc3rnnfghmeknv654
*set*of vertices of*maximal*cardinality that decomposes into a red*and*a blue stable*set*. ... We investigate the complexity of special*and*related cases (e.g., (anti-)chains in partial orders*and*stable matroid bases)*and*exhibit some NP-hard cases as well as polynomial*ones*. ... Acknowledgments We thank anonymous referees*on*an earlier version of this paper for pointing out F. Gavril's work [6] ,*and*Gyula Pap for fruitful conversations. ...##
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Chains, antichains, and fibres

1987
*
Journal of combinatorial theory. Series A
*

Call a subset of an ordered

doi:10.1016/0097-3165(87)90029-x
fatcat:hpiplvqskzbnlcawu37w3pu4vq
*set*a fibre if*it*meets every*maximal**antichain*. ...*One*example is this: In a finite graph G every minimal transuersal is independent if*and*only if G*contains**no*path of length three. !D 1987 Academic ems, hc. ... An ordered*set*is N-free if*its*diagram*contains**no*N, that is, if*it**contains**no*cover-preserving subset isomorphic to (a < c, b < c, b <d). ...##
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On the size of maximal chains and the number of pairwise disjoint maximal antichains

2010
*
Discrete Mathematics
*

in I k , then P

doi:10.1016/j.disc.2010.06.035
fatcat:4uskksdx3nc2rc77sqmfev4oiy
*contains*k pairwise disjoint*maximal**antichains*. ... For each integer k ≥ 3, we find all*maximal*intervals I k of natural numbers with the following property: whenever the number of elements in every*maximal*chain in a finite partially ordered*set*P lies ... Acknowledgements The research was largely performed while the second author was visiting Emory University*on*sabbatical. The second author thanks Emory University for*its*hospitality. ...##
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On Deductive Systems of AC Semantics for Rough Sets
[article]

2016
*
arXiv
*
pre-print

In her paper two different semantics,

arXiv:1610.02634v1
fatcat:rybwgw33azdgdhdp5jracsxuaa
*one*for general rough*sets**and*another for general approximation spaces over quasi-equivalence relations, were developed. ...*Antichain*based semantics for general rough*sets*were introduced recently by the present author. ... Suppose the objects represented by the pairs (a, b)*and*(e, f ) are such that a = e*and*b ⊂ f , then*it*is clear that any*maximal**antichain**containing*(e, f ) cannot*contain*any element from {x : x l ...##
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Two-colouring all two-element maximal antichains

1991
*
Journal of combinatorial theory. Series A
*

We give an example of a finite poset P with

doi:10.1016/0097-3165(91)90009-6
fatcat:26fhjtvp5nbifbhexsj2jcmn7a
*no**one*-element*maximal**antichain**and**containing**no*fibre of size at most JP(/2, thus answering a question of Aigner-Andreae*and*disproving a conjecture of Lone-Rival ... A fibre in a partially ordered*set*P is a subset of P meeting every*maximal**antichain*of P. ... In this case only*one*kind of hypergraph (*maximal*complete subgraph, or*maximal*independent*set*) need be considered, since the*complement*of a perfect graph is perfect. ...##
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On maximal antichains consisting of sets and their complements

1980
*
Journal of combinatorial theory. Series A
*

The integers pi= I(i)XI, i =O, l,..., K, are called parameters of X Clements

doi:10.1016/0097-3165(80)90034-5
fatcat:ar7wlqlqc5fotcymc7bgaaqeou
*and*Gronau 141 studied the*maximal**antichains**containing**no**set**and**its**complement*. ... The opposite concept, c-*antichains*, i.e.,*antichains*consisting of*sets**and*their*complements*, was considered in the case k, = I by Bollob&s [ 11. ...##
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Covering Graphs by Colored Stable Sets

2004
*
Electronic Notes in Discrete Mathematics
*

Let G = (V, R ∪ B) be a multigraph with red

doi:10.1016/j.endm.2004.03.029
fatcat:s3nhdkbwcbgm3nrzuzrwt7zy5e
*and*blue edges. G is an R/B-split graph if V is the union of a red*and*a blue stable*set*. ...*On*the other hand, finding a*maximal*R/B-subgraph is N P-hard already for the class of comparability graphs of series-parallel orders. ... Acknowledgment The authors are grateful to Nguyen Tien Thanh for pointing out the relationship between R/B-split graphs*and*König graphs. ...##
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A note on blockers in posets
[article]

2004
*
arXiv
*
pre-print

The blocker A^* of an

arXiv:math/0403094v1
fatcat:igiqjkk2sfbjrjqwgjfiqi64ru
*antichain*A in a finite poset P is the*set*of elements minimal with the property of having with each member of A a common predecessor. The following is done: 1. ... Connections with the question of finding minimal size blocking*sets*for certain*set*families are discussed. ... A crucial property in this*setting*is that if A is an*antichain*(*no**set**contains*another), then A * * = A. ...##
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Page 4982 of Mathematical Reviews Vol. , Issue 86k
[page]

1986
*
Mathematical Reviews
*

Then (1) P is the union of

*antichain*cutsets if*and*only if*it**contains**no*alternating-cover cycle; (2) z in P is*contained*in an*antichain*cutset if*and*only if there is*no*generalized alternating-cover ... Order 1 (1985),*no*. 3, 235-247. A subset A of an ordered*set*P is called a cutset if each*maximal*chain of P meets A. If, in addition, A is an*antichain*,*it*is called an*antichain*cutset. ...##
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Fibres and ordered set coloring

1991
*
Journal of combinatorial theory. Series A
*

A Jbre F of a partially ordered

doi:10.1016/0097-3165(91)90083-s
fatcat:eimtxh5vt5dabnkxedwhr6tmoq
*set*P is a subset which intersects each nontrivial*maximal**antichain*of P. ... Let I be the smallest constant such that each finite partially ordered*set*P*contains*a fibre of size at most I '1 PI. ... Conversely, a*maximal**antichain*A disjoint from F*contains*exactly*one*of the literals U, ii for each u E U. ...##
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On a new reciprocity, distribution and duality law

1957
*
Pacific Journal of Mathematics
*

All depend

doi:10.2140/pjm.1957.7.1125
fatcat:4z6dxcuwpvfp3gttysuvxrcyk4
*on*the interconnection between*maximal*chains*and**maximal**antichains*of ordered*sets*. By considering ordered*sets**one*achieves considerable generality. ...*On*the other hand, the parallel considerations of*maximal*chains*and**maximal**antichains*of S give rise to a new kind of interconnection of elements of P 1 ! ... Proof Suppose,*on*the contrary, that T*contains*a*maximal*chain M*and*a*maximal**antichain*A so that (10.3) A Γ\ M^v . ...##
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Two forbidden induced subgraphs and well-quasi-ordering

2011
*
Discrete Mathematics
*

Since two graphs G

doi:10.1016/j.disc.2011.04.023
fatcat:bq6wz72lgrfrvpth4xkatbi5am
*and*H are isomorphic if*and*only if their*complements*are isomorphic, we conclude that Claim 2. A*set*of graphs is a WQO if*and*only if the*set*of their*complements*is a WQO. ...*It*is known that a class of graphs defined by a single forbidden induced subgraph G is well-quasi-ordered by the induced subgraph relation if*and*only if G is an induced subgraph of P 4 . ... Introduction A partial order*on*graphs is a well-quasi-order (WQO for short) if*it**contains**no*infinite*antichains*, i.e.*no*infinite*sets*of graphs pairwise incomparable with respect to the order. ...
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