Simplicial elimination schemes, extremal lattices and maximal antichain lattices

Order ◽  
1996 ◽  
Vol 13 (2) ◽  
pp. 159-173 ◽  
Author(s):  
Michel Morvan ◽  
Lhouari Nourine
Keyword(s):  
Maximal Antichain

scholarly journals Minimum sized fibres in distributive lattices

2001 ◽  
Vol 70 (3) ◽  
pp. 337-350 ◽  
Author(s):  
Dwight Duffus ◽  
Bill Sands
Keyword(s):  
Lower Bound ◽  
Distributive Lattice ◽  
Ordered Set ◽  
The Other ◽  
Distributive Lattices ◽  
Maximal Antichain ◽  
Monotone Property ◽  
Fibre Size

AbstractA subset F of an ordered set X is a fibre of X if F intersects every maximal antichain of X. We find a lower bound on the function ƒ (D), the minimum fibre size in the distributive lattice D, in terms of the size of D. In particular, we prove that there is a constant c such that In the process we show that minimum fibre size is a monotone property for a certain class of distributive lattices. This fact depends upon being able to split every maximal antichain of this class of distributive lattices into two parts so that the lattice is the union of the upset of one part and the downset of the other.

scholarly journals Possible behaviours of the reflection ordering of stationary sets

1995 ◽  
Vol 60 (2) ◽  
pp. 534-547 ◽  
Author(s):  
Jiří Witzany
Keyword(s):  
Partial Ordering ◽  
Generic Extension ◽  
Maximal Antichain ◽  
Uncountable Cardinal ◽  
Image Position ◽  
Stationary Subsets ◽  
Almost All

AbstractIf S, T are stationary subsets of a regular uncountable cardinal κ, we say that S reflects fully in T, S < T, if for almost all α ∈ T (except a nonstationary set) S ∩ α stationary in α. This relation is known to be a well-founded partial ordering. We say that a given poset P is realized by the reflection ordering if there is a maximal antichain 〈Xp: p ∈ P〉 of stationary subsets of Reg(κ) so thatWe prove that if , and P is an arbitrary well-founded poset of cardinality ≤ κ+ then there is a generic extension where P is realized by the reflection ordering on κ.

scholarly journals Sperner's Problem for G-Independent Families

2014 ◽  
Vol 24 (3) ◽  
pp. 528-550
Author(s):  
VICTOR FALGAS-RAVRY
Keyword(s):  
Maximal Antichain ◽  
Independent Edge ◽  
Sperner's Theorem ◽  
Independent Families ◽  
Free Sets

Given a graph G, let Q(G) denote the collection of all independent (edge-free) sets of vertices in G. We consider the problem of determining the size of a largest antichain in Q(G). When G is the edgeless graph, this problem is resolved by Sperner's theorem. In this paper, we focus on the case where G is the path of length n − 1, proving that the size of a maximal antichain is of the same order as the size of a largest layer of Q(G).

scholarly journals Another Note on Dilworth's Decomposition Theorem

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Wim Pijls ◽  
Rob Potharst
Keyword(s):  
Decomposition Theorem ◽  
New Method ◽  
Flow Networks ◽  
Maximal Antichain ◽  
Dilworth’S Theorem

This paper proposes a new proof of Dilworth's theorem. The proof is based upon the minflow/maxcut property in flow networks. In relation to this proof, a new method to find both a Dilworth decomposition and a maximal antichain is presented.

Posets with the Maximal Antichain Property

Order ◽  
2009 ◽  
Vol 27 (1) ◽  
pp. 1-8 ◽  
Author(s):  
Bill Sands
Keyword(s):  
Maximal Antichain

scholarly journals Splitting Numbers of Grids

10.37236/1914 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Dwight Duffus ◽  
Bill Sands
Keyword(s):  
Ordered Set ◽  
Ordered Sets ◽  
Maximal Antichain ◽  
P Elements ◽  
The Family ◽  
Splitting Numbers

For a subset $S$ of a finite ordered set $P$, let $$S\!\uparrow\;=\{x\in P:x\ge s \hbox{ for some }s\in S\}\quad\hbox{and} \quad S\!\downarrow\;=\{x\in P:x\le s \hbox{ for some }s\in S\}.$$ For a maximal antichain $A$ of $P$, let $$s(A)=\max_{A=U\cup D}{|U\!\uparrow|+|D\!\downarrow|\over|P|}\ ,$$ the maximum taken over all partitions $U\cup D$ of $A$, and $$s_k(P)=\min_{A\in {\cal A}(P),|A|=k}s(A)$$ where we assume $P$ contains at least one maximal antichain of $k$ elements. Finally, for a class ${\cal C}$ of finite ordered sets, we define $$s_k({\cal C})=\inf_{P\in {\cal C}}s_k(P).$$ Thus $s_k({\cal C})$ is the greatest proportion $r$ satisfying: every $k$-element maximal antichain of a member $P$ of ${\cal C}$ can be "split" into sets $U$ and $D$ so that $U\!\uparrow\cup\; D\!\downarrow$ contains at least $r|P|$ elements. In this paper we determine $s_k({\cal G}_k)$ for all $k\ge 1$, where ${\cal G}_k=\{{\bf k}\times{\bf n}:n\ge k\}$ is the family of all $k$ by $n$ "grids".

scholarly journals Maximal antichains of isomorphic subgraphs of the Rado graph

Filomat ◽  
2015 ◽  
Vol 29 (9) ◽  
pp. 1919-1923 ◽  
Author(s):  
Milos Kurilic ◽  
Petar Markovic
Keyword(s):  
Positive Integer ◽  
Partial Order ◽  
Maximal Antichain ◽  
Chain Complete ◽  
A Chain

If ?R,E? is the Rado graph andR(R) the set of its copies inside R, then ?R(R), ?? is a chain-complete and non-atomic partial order of the size 2x0 . A family A ? R(R) is a maximal antichain in this partial order iff (1) A ? B does not contain a copy of R, for each different A, B ?A and (2) For each S ? R(R) there is A ? A such that A ? S contains a copy of R. We show that the partial order ?R(R), ?? contains maximal antichains of size 2x0, X0 and n, for each positive integer n (thus, of all possible cardinalities, under CH). The results are compared with the corresponding known results concerning the partial order ?[?]?, ??.

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