scholarly journals Venn Diagrams and Symmetric Chain Decompositions in the Boolean Lattice

10.37236/1755 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Jerrold Griggs ◽  
Charles E. Killian ◽  
Carla D. Savage
Keyword(s):  
Boolean Lattice ◽  
Constructive Proof ◽  
Existence Proof ◽  
Chain Decomposition ◽  
Venn Diagrams ◽  
Minimum Number ◽  
Symmetric Chain Decomposition ◽  
Open Question ◽  
Symmetric Chain

We show that symmetric Venn diagrams for $n$ sets exist for every prime $n$, settling an open question. Until this time, $n=11$ was the largest prime for which the existence of such diagrams had been proven, a result of Peter Hamburger. We show that the problem can be reduced to finding a symmetric chain decomposition, satisfying a certain cover property, in a subposet of the Boolean lattice ${\cal B}_n$, and prove that such decompositions exist for all prime $n$. A consequence of the approach is a constructive proof that the quotient poset of ${\cal B}_n$, under the relation "equivalence under rotation", has a symmetric chain decomposition whenever $n$ is prime. We also show how symmetric chain decompositions can be used to construct, for all $n$, monotone Venn diagrams with the minimum number of vertices, giving a simpler existence proof.

scholarly journals Symmetric Chain Decompositions of Products of Posets with Long Chains

10.37236/7124 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Stefan David ◽  
Hunter Spink ◽  
Marius Tiba
Keyword(s):  
Minimal Element ◽  
Difficult Case ◽  
Chain Decomposition ◽  
Graded Poset ◽  
Canonical Bijection ◽  
Symmetric Chain Decomposition ◽  
Symmetric Chain ◽  
Long Chains

We ask if there exists a symmetric chain decomposition of the cuboid $Q_k \times n$ such that no chain is taut, i.e. no chain has a subchain of the form $(a_1,\ldots, a_k,0)\prec \cdots\prec (a_1,\ldots,a_k,n-1)$. In this paper, we show this is true precisely when $k \ge 5$ and $n\ge 3$. This question arises naturally when considering products of symmetric chain decompositions which induce orthogonal chain decompositions — the existence of the decompositions provided in this paper unexpectedly resolves the most difficult case of previous work by the second author on almost orthogonal symmetric chain decompositions (2017), making progress on a conjecture of Shearer and Kleitman (1979). In general, we show that for a finite graded poset $P$, there exists a canonical bijection between symmetric chain decompositions of $P \times m$ and $P \times n$ for $m, n\ge rk(P) + 1$, that preserves the existence of taut chains. If $P$ has a unique maximal and minimal element, then we also produce a canonical $(rk(P) +1)$ to $1$ surjection from symmetric chain decompositions of $P \times (rk(P) + 1)$ to symmetric chain decompositions of $P \times rk(P)$ which sends decompositions with taut chains to decompositions with taut chains.

scholarly journals Spherical Venn Diagrams with Involutory Isometries

10.37236/678 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Frank Ruskey ◽  
Mark Weston
Keyword(s):  
Boolean Lattice ◽  
Venn Diagram ◽  
Antipodal Point ◽  
Chain Decomposition ◽  
Venn Diagrams ◽  
Reflective Symmetry

In this paper we give a construction, for any $n$, of an $n$-Venn diagram on the sphere that has antipodal symmetry; that is, the diagram is fixed by the map that takes a point on the sphere to the corresponding antipodal point. Thus, along with certain diagrams due to Anthony Edwards which can be drawn with rotational and reflective symmetry, for any isometry of the sphere that is an involution, there exists an $n$-Venn diagram on the sphere invariant under that involution. Our construction uses a recursively defined chain decomposition of the Boolean lattice.

Applications of the symmetric chain decomposition of the lattice of divisors

Order ◽  
1994 ◽  
Vol 11 (1) ◽  
pp. 41-46
Author(s):  
Jerrold R. Griggs ◽  
Chuanzhong Zhu
Keyword(s):  
Chain Decomposition ◽  
Symmetric Chain Decomposition ◽  
Symmetric Chain

scholarly journals Symmetric Chain Decomposition of Necklace Posets

10.37236/1178 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Vivek Dhand
Keyword(s):  
Disjoint Union ◽  
Cyclic Permutation ◽  
Chain Decomposition ◽  
Chain Order ◽  
Symmetric Chain Decomposition ◽  
Symmetric Chain ◽  
Symmetric Chain Order

A finite ranked poset is called a symmetric chain order if it can be written as a disjoint union of rank-symmetric, saturated chains. If $\mathcal{P}$ is any symmetric chain order, we prove that $\mathcal{P}^n/\mathbb{Z}_n$ is also a symmetric chain order, where $\mathbb{Z}_n$ acts on $\mathcal{P}^n$ by cyclic permutation of the factors.

scholarly journals A Decomposition of Parking Functions by Undesired Spaces

10.37236/5940 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Melody Bruce ◽  
Michael Dougherty ◽  
Max Hlavacek ◽  
Ryo Kudo ◽  
Ian Nicolas
Keyword(s):  
Parking Functions ◽  
Partition Lattice ◽  
Self Duality ◽  
Noncrossing Partition ◽  
Symmetric Chain Decomposition ◽  
Order Complexes ◽  
Maximal Chains ◽  
Symmetric Chain ◽  
Noncrossing Partition Lattice

There is a well-known bijection between parking functions of a fixed length and maximal chains of the noncrossing partition lattice which we can use to associate to each set of parking functions a poset whose Hasse diagram is the union of the corresponding maximal chains. We introduce a decomposition of parking functions based on the largest number omitted and prove several theorems about the corresponding posets. In particular, they share properties with the noncrossing partition lattice such as local self-duality, a nice characterization of intervals, a readily computable Möbius function, and a symmetric chain decomposition. We also explore connections with order complexes, labeled Dyck paths, and rooted forests.

scholarly journals On Orthogonal Symmetric Chain Decompositions

10.37236/8531 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Karl Däubel ◽  
Sven Jäger ◽  
Torsten Mütze ◽  
Manfred Scheucher
Keyword(s):  
Recent Result ◽  
Single Element ◽  
Edge Disjoint ◽  
Minimum Number ◽  
Orthogonal Decompositions ◽  
Symmetric Chain

The $n$-cube is the poset obtained by ordering all subsets of $\{1,\ldots,n\}$ by inclusion, and it can be partitioned into $\binom{n}{\lfloor n/2\rfloor}$ chains, which is the minimum possible number. Two such decompositions of the $n$-cube are called orthogonal if any two chains of the decompositions share at most a single element. Shearer and Kleitman conjectured in 1979 that the $n$-cube has $\lfloor n/2\rfloor+1$ pairwise orthogonal decompositions into the minimum number of chains, and they constructed two such decompositions. Spink recently improved this by showing that the $n$-cube has three pairwise orthogonal chain decompositions for $n\geq 24$. In this paper, we construct four pairwise orthogonal chain decompositions of the $n$-cube for $n\geq 60$. We also construct five pairwise edge-disjoint symmetric chain decompositions of the $n$-cube for $n\geq 90$, where edge-disjointness is a slightly weaker notion than orthogonality, improving on a recent result by Gregor, Jäger, Mütze, Sawada, and Wille.  

scholarly journals Filling of closed surfaces

2018 ◽  
Vol 10 (04) ◽  
pp. 897-913 ◽  
Author(s):  
Bidyut Sanki
Keyword(s):  
Disjoint Union ◽  
Mapping Class ◽  
Constructive Proof ◽  
Closed Curves ◽  
Closed Surfaces ◽  
Minimal Filling ◽  
Minimum Number ◽  
Genus Formula ◽  
Positive Integers ◽  
Closed Oriented Surface

Let [Formula: see text] denote a closed oriented surface of genus [Formula: see text]. A set of simple closed curves is called a filling of [Formula: see text] if its complement is a disjoint union of discs. The mapping class group [Formula: see text] of genus [Formula: see text] acts on the set of fillings of [Formula: see text]. The union of the curves in a filling forms a graph on the surface which is a so-called decorated fat graph. It is a fact that two fillings of [Formula: see text] are in the same [Formula: see text]-orbit if and only if the corresponding fat graphs are isomorphic. We prove that any filling of [Formula: see text] whose complement is a single disc (i.e. a so-called minimal filling) has either three or four closed curves and in each of these two cases, there is a unique such filling up to the action of [Formula: see text]. We provide a constructive proof to show that the minimum number of discs in the complement of a filling pair of [Formula: see text] is two. Finally, given positive integers [Formula: see text] and [Formula: see text] with [Formula: see text], we construct a filling pair of [Formula: see text] such that the complement is a union of [Formula: see text] topological discs.

scholarly journals On the existence of symmetric chain decompositions in a quotient of the Boolean lattice

2009 ◽  
Vol 309 (17) ◽  
pp. 5278-5283 ◽  
Author(s):  
Zongliang Jiang ◽  
Carla D. Savage
Keyword(s):  
Boolean Lattice ◽  
Symmetric Chain
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