Polygons with a (P, 1)-stable theory

Polygons with a $(P,1)$-stable theory are considered. A criterion of being $(P,1)$-stable for a polygon is established. As a consequence of the main criterion we prove that a polygon $_SS$, where $S$ is a group, is $(P,1)$-stable if and only if $S$ is a finite group. It is shown that the class of all polygons with monoid $S$ is $(P,1)$-stable only if $S$ is a one-element monoid. $(P,1)$-stability criteria are presented for polygons over right and left zero monoids.