Minimal null designs and a density theorem of posets
type: Item
instance of: scholarly article
title: Minimal null designs and a density theorem of posets
zbMATH Open document ID: 0908.05024
author: Soojin Cho
published in: European Journal of Combinatorics
publication date: 1999-03-22T00:00:00Z
review text: Let \(P\) be a finite, ranked poset with highest rank \(n\) and let \(P_i\) denote the set of elements of rank \(i\). For a finite set \(X\), let \(\mathbb{R}[X]\) denote the real vector space of linear combinations of the elements of \(X\). For integers \(0\leq t<k\leq n\), define the incidence map \(d_{k,t}: \mathbb{R}[P_k]\to \mathbb{R}[P_t]\) as the extension by linearity where \(d_{k,t}(x)= \sum_{y\leq x,y\in P_t}y\), for each \(x\in P_k\). For \(0\leq t<k\leq n\), \(N_p(t, k)\), the space of null \((t,k)\)-designs for \(P\), is the kernel of \(d_{k,t}\). When \(P\) is the lattice of subsets of an \(n\)-set, these are the usual null designs. This paper reproves many of the standard results and proves some new results in this more general setting. It also gives some new characterizations of null designs for the Boolean lattice and the generalized Boolean lattice of \(\{0,1,\dots, q\}\)-valued functions on an \(n\)-set.
Mathematics Subject Classification ID: 05B30
zbMATH DE Number: 1199924
zbMATH Keywords: ranked poset
zbMATH Keywords: lattice of subsets
zbMATH Keywords: null designs
zbMATH Keywords: Boolean lattice
reviewed by: Jack E. Graver
MaRDI profile type: Publication
full work available at URL: eujc.1997.0201
OpenAlex ID: W2024020678
DOI: 10.1006/EUJC.1997.0201
description: scientific article; zbMATH DE number 1199924