Primitive Connected and Additive Theories of Polygons

We study into monoids $S$ the class of all $S$-polygons over which is primitive normal, primitive connected, or additive, that is, the monoids $S$ the theory of any $S$-polygon over which is primitive normal, primitive connected, or additive. It is proved that the class of all $S$-polygons is primitive normal iff $S$ is a linearly ordered monoid, and that it is primitive connected iff $S$ is a group. It is pointed out that there exists no monoid $S$ with an additive class of all $S$-polygons.