Publication

Packing the Boolean lattice with copies of a poset

Istvan Tomon
2020
Journal Articles

Abstract

The Boolean lattice (2[n],subset of) is the family of all subsets of [n]={1,MIDLINE HORIZONTAL ELLIPSIS,n}, ordered by inclusion. Let P be a partially ordered set. We prove that if n is sufficiently large, then there exists a packing P of copies of P in (2[n],subset of) that covers almost every element of 2[n]: P might not cover the minimum and maximum of 2[n], and at most |P|-1 additional points due to divisibility. In particular, if |P| divides 2n-2, then the truncated Boolean lattice 2[n]-{ null ,[n]} can be partitioned into copies of P. This confirms a conjecture of Lonc from 1991.

Official source

https://infoscience.epfl.ch/entities/publication/b137e2ca-8811-4e9f-8011-4458c4411ba0

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Istvan Tomon
2020
Journal Articles

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/b137e2ca-8811-4e9f-8011-4458c4411ba0

About this result

Related concepts (32)

Partially ordered set

In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other. The word partial is used to indicate that not every pair of elements needs to be comparable; that is, there may be pairs for which neither element precedes the other. Partial orders thus generalize total orders, in which every pair is comparable. Formally, a partial order is a homogeneous binary relation that is reflexive, transitive and antisymmetric.

Subset

In mathematics, set A is a subset of a set B if all elements of A are also elements of B; B is then a superset of A. It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B. The relationship of one set being a subset of another is called inclusion (or sometimes containment). A is a subset of B may also be expressed as B includes (or contains) A or A is included (or contained) in B. A k-subset is a subset with k elements. The subset relation defines a partial order on sets.

Partially ordered group

In abstract algebra, a partially ordered group is a group (G, +) equipped with a partial order "≤" that is translation-invariant; in other words, "≤" has the property that, for all a, b, and g in G, if a ≤ b then a + g ≤ b + g and g + a ≤ g + b. An element x of G is called positive if 0 ≤ x. The set of elements 0 ≤ x is often denoted with G+, and is called the positive cone of G. By translation invariance, we have a ≤ b if and only if 0 ≤ -a + b.

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