Polygons with primitive normal and additive theories

Previously, primitive normal, primitive connected, and additive theories of $S$-polygons were studied. In particular, it was proved that the class of all $S$-polygons is primitive normal iff $S$ is a linearly ordered monoid. The present paper is a continuation of this research. Here, Spolygons with primitive normal, additive, and antiadditive theories are described in the language of a primitive equivalence structure. It is shown that the class of all $S$-polygons is antiadditive only for a linearly ordered monoid $S$, that is, this class is antiadditive iff it is primitive normal.