Minimal null designs and a density theorem of posets
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.
scientific article; zbMATH DE number 1199924