Cube orders (Q1970927)

Let \(P= (X,\prec)\) be a finite poset, i.e., \(X\) is a finite set and \(\prec\) is an irreflexive and transitive binary relation on \(X\), and let \({\mathcal P}\) be the class of all finite posets. If \({\mathcal S}\) is a family of geometric objects in the \(m\)-dimensional Euclidean space \(R^m\), then \(P\in {\mathcal P}\) is called \({\mathcal S}\)-representable if there is a mapping \(f\) from \(X\) into \({\mathcal S}\) such that \(x\prec y\) iff \(f(x) \subset f(y)\), for each \(x,y\in X\). The author focuses on \({\mathcal S}\) families of box-like polyhedra in \(R^n\) with special attention to poset dimensionality. Especially, he describes \({\mathcal S}\)-representable members of \({\mathcal P}\) for \({\mathcal S}={\mathcal C}_m\), where \({\mathcal C}_m\) is the family of all cubes with edges parallel to the axes in \(R^m\), and for \({\mathcal S}={\mathcal C}^*\), where \({\mathcal C}_m^*\) is the family of all cubes in \(R^m\).