Angle orders, regular n-gon orders and the crossing number
Let \(X:=\{x_ 1,...,x_ m\}\) be a finite set, \(<\) be a partial order relation on X. A family \(\Phi =\{S_ 1,...,S_ m\}\) of sets is called a representation of \((X,<)\) iff \((x_ i<x_ j\) iff \(S_ i\subset S_ j\) for all \(i\neq j)\). \((X,<)\) is called angle order (regular n-gon order, circle order) iff there exists a representation \(\Phi\) consisting of angular regions (of regular polygons having n sides, circles) in the real affine plane. Let \(P:=(X,<)\) be a partial order, and let dim P denote the smallest number of linear orders whose intersection yields P. Using the concept of crossing number of a partial order defined by \textit{M. C. Golumbic}, \textit{D. Rotem}, and \textit{J. Urrutia} [Discrete Math. 43, 37-46 (1983; Zbl 0502.05050)], the authors prove: There exists P such that dim P\(=4\), \(| X| =14\), and for no value of n this partial order P is a regular n-gon order. On the contrary, dim P\(=3\) implies that P is a regular n-gon order for any \(n\geq 3\), where in particular, the triangle representation can be chosen equilateral. Conversely, if P is a regular n-gon oder, then dim \(P\leq n\), the bound being reached by some suitable P for any \(n\geq 3\). Finally, the authors consider the following question from \textit{P. C. Fishburn} and \textit{W. T. Trotter} [Order 1, 333-343 (1985; Zbl 0558.06003)]: What is the least dimension of a partial order which is not an angle order ? They give an example of a non-angle order P such that dim P\(=6\), \(| X| =64\).
scientific article; zbMATH DE number 4047783