Boolean Lattice Research Articles

Poset Ramsey Numbers for Boolean Lattices

For each positive integer n, let Qn denote the Boolean lattice of dimension n. For posets \(P, P^{\prime }\), define the poset Ramsey number \(R(P,P^{\prime })\) to be the least N such that for any red/blue coloring of the elements of QN, there exists either a subposet isomorphic to P with all elements red, or a subposet isomorphic to \(P^{\prime }\) with all elements blue. Axenovich and Walzer introduced this concept in Order (2017), where they proved R(Q2,Qn) ≤ 2n + 2 and R(Qn,Qm) ≤ mn + n + m. They later proved 2n ≤ R(Qn,Qn) ≤ n2 + 2n. Walzer later proved R(Qn,Qn) ≤ n2 + 1. We provide some improved bounds for R(Qn,Qm) for various \(n,m \in \mathbb {N}\). In particular, we prove that R(Qn,Qn) ≤ n2 − n + 2, \(R(Q_{2}, Q_{n}) \le \frac {5}{3}n + 2\), and \(R(Q_{3}, Q_{n}) \le \lceil \frac {37}{16}n + \frac {55}{16}\rceil \). We also prove that R(Q2,Q3) = 5, and \(R(Q_{m}, Q_{n}) \le \left \lceil \left (m - 1 + \frac {2}{m+1} \right )n + \frac {1}{3} m + 2\right \rceil \) for all n > m ≥ 4.

Finite distributive nearlattices

Our main goal is to develop a representation for finite distributive nearlattices through certain ordered structures. This representation generalizes the well-known representation given by Birkhoff for finite distributive lattices through finite posets. We also study finite distributive nearlattices through the concepts of dual atoms, boolean elements, complemented elements and irreducible elements. We prove that the sets of boolean elements and complemented elements form semi-boolean algebras. We show that the set of boolean elements of a finite distributive lattice is a boolean lattice.

Explicit Δ-edge-coloring of consecutive levels in a divisor lattice

We show that the explicit 1-factorizations of the middle levels in a Boolean lattice, defined by Duffus et al. (1994) [2], and by Kierstead and Trotter (1988) [7], can both be generalized in a simple way to define explicit Δ-edge-colorings of any two consecutive levels in any divisor lattice.

A numeral system for the middle-levels graphs

A sequence S of restricted-growth strings unifies the presentation of middle-levels graphs M k as follows, for 0 < k ∈ Z . Recall M k is the subgraph in the Hasse diagram of the Boolean lattice 2 [2 k +1] induced by the k - and ( k +1)-levels. The dihedral group D 4 k +2 acts on M k via translations mod (2 k + 1) and complemented reversals.The first (2 k )!/( k !( k +1)!) terms of S stand for the orbits of V ( M k ) under such D 4 k +2 -action, via the lexical matching colors 0, 1, ... , k on the k +1 edges at each vertex. So, S is proposed here as a convenient numeral system for the graphs M k . Color 0 allows to reorder S via an integer sequence that behaves as an idempotent permutation on its first (2 k )!/( k !( k +1)!) terms, for each 0 < k ∈ Z . Related properties hold for the remaining colors 1, ... , k .

Largest Family Without a Pair of Posets on Consecutive Levels of the Boolean Lattice

Suppose k ≥ 2 is an integer. Let Yk be the poset with elements x1,x2,y1,y2,…,yk− 1 such that y1 < y2 < ⋯ < yk− 1 < x1,x2 and let Y_{k}^{prime } be the same poset but all relations reversed. We say that a family of subsets of [n] contains a copy of Yk on consecutive levels if it contains k + 1 subsets F1,F2,G1,G2,…,Gk− 1 such that G1 ⊂ G2 ⊂⋯ ⊂ Gk− 1 ⊂ F1,F2 and |F1| = |F2| = |Gk− 1| + 1 = |Gk− 2| + 2 = ⋯ = |G1| + k − 1. If both Yk and Y^{prime }_{k} on consecutive levels are forbidden, the size of the largest such family is denoted by text {La}_{mathrm {c}}left (n, Y_{k}, Y^{prime }_{k}right ). In this paper, we will determine the exact value of text {La}_{mathrm {c}}left (n, Y_{k}, Y^{prime }_{k}right ).

De Finetti Lattices and Magog Triangles

The order ideal $B_{n,2}$ of the Boolean lattice $B_n$ consists of all subsets of size at most $2$. Let $F_{n,2}$ denote the poset refinement of $B_{n,2}$ induced by the rules: $i &lt; j$ implies $\{i \} \prec \{ j \}$ and $\{i,k \} \prec \{j,k\}$. We give an elementary bijection from the set $\mathcal{F}_{n,2}$ of linear extensions of $F_{n,2}$ to the set of shifted standard Young tableau of shape $(n, n-1, \ldots, 1)$, which are counted by the strict-sense ballot numbers. We find a more surprising result when considering the set $\mathcal{F}_{n,2}^{1}$ of minimal poset refinements in which each singleton is comparable with all of the doubletons. We show that $\mathcal{F}_{n,2}^{1}$ is in bijection with magog triangles, and therefore is equinumerous with alternating sign matrices. We adopt our proof techniques to show that row reversal of an alternating sign matrix corresponds to a natural involution on gog triangles.

Minimum Weight Flat Antichains of Subsets

Building on classical theorems of Sperner and Kruskal-Katona, we investigate antichains $\mathcal F$ in the Boolean lattice $B_n$ of all subsets of $[n]:=\{1,2,\dots,n\}$, where $\mathcal F$ is flat, meaning that it contains sets of at most two consecutive sizes, say $\mathcal F=\mathcal{A}\cup\mathcal{B}$, where $\mathcal{A}$ contains only $k$-subsets, while $\mathcal{B}$ contains only $(k-1)$-subsets. Moreover, we assume $\mathcal{A}$ consists of the first $m$ $k$-subsets in squashed (colexicographic) order, while $\mathcal{B}$ consists of all $(k-1)$-subsets not contained in the subsets in $\mathcal{A}$. Given reals $\alpha,\beta>0$, we say the weight of $\mathcal F$ is $\alpha\cdot|\mathcal{A}|+\beta\cdot|\mathcal{B}|$. We characterize the minimum weight antichains $\mathcal F$ for any given $n,k,\alpha,\beta$, and we do the same when in addition $\mathcal F$ is a maximal antichain. We can then derive asymptotic results on both the minimum size and the minimum Lubell function.

Computing Linear Extensions for Polynomial Posets Subject to Algebraic Constraints

In this paper we consider the classical problem of computing linear extensions of a given poset which is well known to be a difficult problem. However, in our setting the elements of the poset are multivariate polynomials, and only a small "admissible" subset of these linear extensions, determined implicitly by the evaluation map, are of interest. This seemingly novel problem arises in the study of global dynamics of gene regulatory networks in which case the poset is a Boolean lattice. We provide an algorithm for solving this problem using linear programming for arbitrary partial orders of linear polynomials. This algorithm exploits this additional algebraic structure inherited from the polynomials to efficiently compute the admissible linear extensions. The biologically relevant problem involves multilinear polynomials and we provide a construction for embedding it into an instance of the linear problem.

The Boolean Rainbow Ramsey Number of Antichains, Boolean Posets and Chains

Motivated by the paper, Boolean lattices: Ramsey properties and embeddings Order, 34 (2) (2017), of Axenovich and Walzer, we study the Ramsey-type problems on the Boolean lattices. Given posets $P$ and $Q$, we look for the smallest Boolean lattice $\mathcal{B}_N$ such that any coloring of elements of $\mathcal{B}_N$ must contain a monochromatic $P$ or a rainbow $Q$ as an induced subposet. This number $N$ is called the Boolean rainbow Ramsey number of $P$ and $Q$ in the paper.&#x0D; Particularly, we determine the exact values of the Boolean rainbow Ramsey number for $P$ and $Q$ being the antichains, the Boolean posets, or the chains. From these results, we also derive some general upper and lower bounds of the Boolean rainbow Ramsey number for general $P$ and $Q$ in terms of the poset parameters.

Independent sets in the middle two layers of Boolean lattice

For an odd integer n=2d−1, let B(n,d) be the subgraph of the hypercube Qn induced by the two largest layers. In this paper, we describe the typical structure of independent sets in B(n,d) and give precise asymptotics on the number of them. The proofs use Sapozhenko's graph container method and a recently developed method of Jenssen and Perkins, which combines Sapozhenko's graph container lemma with the cluster expansion for polymer models from statistical physics.

Pairwise coexistence of effects versus coexistence

The concept of coexistence of quantum mechanical effects is reviewed. We distinguish between coexistence and pairwise coexistence and give an example showing the non-equivalence the two concepts. A theorem on some closures of sets of pairwise coexistent effects is proved. Some proofs of the well-known fact that any set of pairwise coexistent (mutually compatible) sharp effects is coexistent are considered. In particular, a proof is presented that is based on the statement that every countably complete Boolean lattice of sharp effects is closed in some topology.

Families in posets minimizing the number of comparable pairs

AbstractGiven a graded poset we say a family is centered if it is obtained by ‘taking sets as close to the middle layer as possible.’ A poset is said to have the centeredness property if for any , among all families of size in , centered families contain the minimum number of comparable pairs. Kleitman showed that the Boolean lattice has the centeredness property. It was conjectured by Noel, Scott, and Sudakov, and by Balogh and Wagner, that the poset also has the centeredness property, provided is sufficiently large compared with . We show that this conjecture is false for all and investigate the range of for which it holds. Further, we improve a result of Noel, Scott, and Sudakov by showing that the poset of subspaces of has the centeredness property. Several open questions are also given.

Improved Bounds for Induced Poset Saturation

Given a finite poset $\mathcal{P}$, a family $\mathcal{F}$ of elements in the Boolean lattice is induced-$\mathcal{P}$-saturated if $\mathcal{F}$ contains no copy of $\mathcal{P}$ as an induced subposet but every proper superset of $\mathcal{F}$ contains a copy of $\mathcal{P}$ as an induced subposet. The minimum size of an induced-$\mathcal{P}$-saturated family in the $n$-dimensional Boolean lattice, denoted $\mathrm{sat}^*(n,\mathcal{P})$, was first studied by Ferrara et al. (2017).&#x0D; Our work focuses on strengthening lower bounds. For the 4-point poset known as the diamond, we prove $\mathrm{sat}^*(n,\Diamond)\geq\sqrt{n}$, improving upon a logarithmic lower bound. For the antichain with $k+1$ elements, we prove $$\mathrm{sat}^*(n,\mathcal{A}_{k+1})\geq \left(1-\frac{1}{\log_2k}\right)\frac{kn}{\log_2 k}$$ for $n$ sufficiently large, improving upon a lower bound of $3n-1$ for $k\geq 3$.

An Efficient, Parallelized Algorithm for Optimal Conditional Entropy-Based Feature Selection.

In Machine Learning, feature selection is an important step in classifier design. It consists of finding a subset of features that is optimum for a given cost function. One possibility to solve feature selection is to organize all possible feature subsets into a Boolean lattice and to exploit the fact that the costs of chains in that lattice describe U-shaped curves. Minimization of such cost function is known as the U-curve problem. Recently, a study proposed U-Curve Search (UCS), an optimal algorithm for that problem, which was successfully used for feature selection. However, despite of the algorithm optimality, the UCS required time in computational assays was exponential on the number of features. Here, we report that such scalability issue arises due to the fact that the U-curve problem is NP-hard. In the sequence, we introduce the Parallel U-Curve Search (PUCS), a new algorithm for the U-curve problem. In PUCS, we present a novel way to partition the search space into smaller Boolean lattices, thus rendering the algorithm highly parallelizable. We also provide computational assays with both synthetic data and Machine Learning datasets, where the PUCS performance was assessed against UCS and other golden standard algorithms in feature selection.

Multicolor chain avoidance in the Boolean lattice

Given a collection of colored chain posets, we estimate the number of colored subsets of the Boolean lattice which avoid all chains in the collection. We do this by applying the hypergraph container lemma recursively, with different uniformities at each stage, using a balanced supersaturation result for a certain non-uniform hypergraph encoding forbidden configurations.

Solution sets of systems of equations over finite lattices and semilattices

Solution sets of systems of homogeneous linear equations over fields are characterized as being subspaces, i.e., sets that are closed under linear combinations. Our goal is to characterize solution sets of systems of equations over arbitrary finite algebras by a similar closure condition. We show that solution sets are always closed under the centralizer of the clone of term operations of the given algebra; moreover, the centralizer is the only clone that could characterize solution sets. If every centralizer-closed set is the set of all solutions of a system of equations over a finite algebra, then we say that the algebra has Property {{,mathrm{(SDC)},}}. Our main result is the description of finite lattices and semilattices with Property {{,mathrm{(SDC)},}}: we prove that a finite lattice has Property {{,mathrm{(SDC)},}} if and only if it is a Boolean lattice, and a finite semilattice has Property {{,mathrm{(SDC)},}} if and only if it is distributive.

Boolean FIP ring extensions

We characterize extensions of commutative rings whose sets of subextensions [R, S] are finite (i.e. has the FIP property) and are Boolean lattices, that we call Boolean FIP extensions. Some characterizations involve “factorial” properties of the poset [R, S]. A non-trivial result is that each subextension of a Boolean FIP extension is simple (i.e. is a simple pair).

Boolean lattices in finite alternating and symmetric groups

Abstract Given a group G and a subgroup H, we let $\mathcal {O}_G(H)$ denote the lattice of subgroups of G containing H. This article provides a classification of the subgroups H of G such that $\mathcal {O}_{G}(H)$ is Boolean of rank at least $3$ when G is a finite alternating or symmetric group. Besides some sporadic examples and some twisted versions, there are two different types of such lattices. One type arises by taking stabilisers of chains of regular partitions, and the other arises by taking stabilisers of chains of regular product structures. As an application, we prove in this case a conjecture on Boolean overgroup lattices related to the dual Ore’s theorem and to a problem of Kenneth Brown.

On difference graphs and the local dimension of posets

The dimension of a partially-ordered set (poset), introduced by Dushnik and Miller (1941), has been studied extensively in the literature. Recently, Ueckerdt (2016) proposed a variation called local dimension which makes use of partial linear extensions. While local dimension is bounded above by dimension, they can be arbitrarily far apart as the dimension of the standard example is n while its local dimension is only 3. Hiraguchi (1955) proved that the maximum dimension of a poset of order n is n∕2. However, we find a very different result for local dimension, proving a bound of Θ(n∕logn). This follows from connections with covering graphs using difference graphs which are bipartite graphs whose vertices in a single class have nested neighborhoods. We also prove that the local dimension of the n-dimensional Boolean lattice is Ω(n∕logn) and make progress toward resolving an analogue of the removable pair conjecture for local dimension.

Big lattices of module classes induced by preradicals

We introduce new big lattices of classes of R-modules induced by suitable preradicals, the big lattice of $$\sigma $$-hereditary classes, the big lattice of$$\sigma $$-cohereditary classes, and the big lattice of $$\sigma $$-open classes. We prove that their Skeletons are boolean lattices which generalize and extend the lattice of natural classes, the lattice of conatural classes and the Skeleton of the lattice of hereditary torsion theories, respectively. We also introduce the lattice of$$\sigma $$-hereditary torsion theories, for a exact preradical $$\sigma $$. In case that $$\sigma $$ be the identity preradical, we obtain the usual lattice of hereditary torsion classes.