Related papers: Boolean dimension of a Boolean lattice

For a partially ordered set P, let Co(P) denote the lattice of all order-convex subsets of P. For a positive integer n, we denote by SUB(LO) (resp., SUB(n)) the class of all lattices that can be embedded into a product of lattices of convex…

We investigate the structure of the Medvedev lattice as a partial order. We prove that every interval in the lattice is either finite, in which case it is isomorphic to a finite Boolean algebra, or contains an antichain of size…

In general, representations of interval orders may use an arbitrary set of interval lengths. We can define subclasses of interval orders by restricting the allowable lengths of intervals. Motivated by a recent paper of Keller, Trenk, and…

We introduce two variants of the poset saturation problem. For a poset $P$ and the Boolean lattice $\mathcal{B}_n$, a family $\mathcal{F}$ of sets, not necessarily from $\mathcal{B}_n$, is \textit{projective $P$-saturated} if (i) it does…

We characterize the finite distributive lattices which admit a complete valuation, that is bijective over a set of consecutive natural numbers, with the additional conditions of completeness (Definition 2.3). We prove that such lattices are…

We introduce so-called consistent posets which are bounded posets with an antitone involution ' where the lower cones of x,x' and of y,y' coincide provided x,y are different form 0,1 and, moreover, if x,y are different form 0 then their…

For a distributive join-semilattice S with zero, a S-valued poset measure on a poset P is a map m:PxP->S such that m(x,z) <= m(x,y)vm(y,z), and x <= y implies that m(x,y)=0, for all x,y,z in P. In relation with congruence lattice…

In this paper, we first show that the collection of all subsets of \( \mathbb{R} \) having lower dimension \( \gamma \in [0,1] \) is dense in \( \Pi(\mathbb{R}) \), the space of compact subsets of \( \mathbb{R} \). Furthermore, we show that…

We prove that every planar poset $P$ of height $h$ has dimension at most $192h + 96$. This improves on previous exponential bounds and is best possible up to a constant factor. We complement this result with a construction of planar posets…

The Dushnik--Miller dimension of a poset $P$ is the least $d$ for which $P$ can be embedded into a product of $d$ chains. Lewis and Souza showed that the dimension of the divisibility order on the interval of integers $[N/\kappa, N]$ is…

For a positive integer n, we denote by SUB (resp., SUBn) the class of all lattices that can be embedded into the lattice Co(P) of all order-convex subsets of a partially ordered set P (resp., P of length at most n). We prove the following…

Dualization of a monotone Boolean function on a finite lattice can be represented by transforming the set of its minimal 1 to the set of its maximal 0 values. In this paper we consider finite lattices given by ordered sets of their meet and…

We prove that the Fourier dimension of any Boolean function with Fourier sparsity $s$ is at most $O\left(s^{2/3}\right)$. Our proof method yields an improved bound of $\widetilde{O}(\sqrt{s})$ assuming a conjecture of…

A careful study is made of embeddings of posets which have a convex range. We observe that such embeddings share nice properties with the homomorphisms of more restrictive categories; for example, we show that every order embedding between…

Suppose $k \ge 2$ is an integer. Let $Y_k$ be the poset with elements $x_1, x_2, y_1, y_2, \ldots, y_{k-1}$ such that $y_1 < y_2 < \cdots < y_{k-1} < x_1, x_2$ and let $Y_k'$ be the same poset but all relations reversed. We say that a…

The main aim of this article is to give an exposition of the diagrammatic proof due to Boos and Rost of the theorem of Hurwitz that the dimension of a composition algebra is one of 0,1,2,4,8.

The breadth of a Lie algebra $L$ is defined to be the maximal dimension of the image of $ad_x=[x,-]:L\to L$, for $x\in L$. Here, we initiate an investigation into the breadth of three families of Lie algebras defined by posets and provide…

In the area of forbidden subposet problems we look for the largest possible size $La(n,P)$ of a family $\mathcal{F}\subseteq 2^{[n]}$ that does not contain a forbidden inclusion pattern described by $P$. The main conjecture of the area…

We prove that all Euclidean lattices of dimension $n\le 9$ which are generated by their minimal vectors, also possess a basis of minimal vectors. By providing a new counterexample, we show that this is not the case for all dimensions $n\ge…

Let $m$ and $n$ be cardinals with $3\leq m,n\leq\omega$. We show that the class of posets that can be embedded into a distributive lattice via a map preserving all existing meets and joins with cardinalities strictly less than $m$ and $n$…