Circle orders and angle orders (Q908945)

The relation between two partial orders - circle order and angle order - is considered. The finite poset \((X,<)\) is called a circle order, if \[ (1)\quad \forall x,y\in X,\quad x<y\quad \Leftrightarrow \quad f(x)\subset f(y), \] where f is a mapping from X into the set of all circular disks in the Euclidean plane. The poset \((X,<)\) is an angle order, when condition (1) holds for f a mapping from X into the set of all angular regions in the Euclidean plane, each of which consists of all points on rays emanating from a vertex v clockwise from an initial ray \(r_ 1\) to a terminal ray \(r_ 2.\) The question whether every finite circle order is an angle order is discussed in this paper. It is proved in the negative: Some finite circle orders are not angle orders. A detailed proof of this fact is presented.