Triangle, parallelogram, and trapezoid orders
Let \(B_1\) and \(B_2\) be two horizontal lines with \(B_1\) below \(B_2\). Let \(R\) be a finite set of trapezoids where each trapezoid has one base in \(B_1\) and another in \(B_2\). Trapezoids where the baselines have length zero are allowed. For trapezoids \(x,y\in R\) let \(x<y\) iff \(x\) and \(y\) are disjoint, and the base of \(x\) on \(B_1\) is to the left of the baseline of \(y\) on \(B_1\). In this way \((R,\leq)\) becomes a partially ordered set (poset). A poset \((P,\leq)\) is called a \textit{trapezoid order} if there is a set \(R\) of trapezoids such that \(P\cong R\) as posets. The poset \(P\) is a \textit{parallelogram order} (resp.~\textit{triangle order}) if each of the trapezoids in \(R\) are parallelograms (resp.~triangles.) If each of the geometric objects in \(R\) (trapezoids, parallelograms, triangles) have a unit area, then \(P\) is a \textit{unit} order. If none of the geometric objects in \(R\) properly contains another object from \(R\), then \(P\) is a \textit{proper} order. A trapezoid order \(P\) which is not necessarily unit or proper is a \textit{plain} order. Out of the 20 nontrivial combinations of a trapezoid order \(P\) being unit/non-unit, proper/non-proper, plain/non-plain and parallelogram/triangle/trapezoid, explicit examples are given that show that all these 20 classes are distinct. Finally, a triangle order \(P\) is an \textit{up-triangle order} if each triangle in \(R\) has its zero length base in \(B_2\). In a short argument, it is shown that every up-triangle order is a unit trapezoid order, and it is left as an open problem to classify the up-triangle orders within the sub-lattice of classes of trapezoid orders.
scientific article; zbMATH DE number 5729755