$P$-stable polygons

$P$-stable polygons are studied. It is proved that the property of being $(P,s)$-, $(P,a)$-, and $(P,e)$-stable for the class of all polygons over a monoid $S$ is equivalent to $S$ being a group. We describe the structure of $(P,s)$-, $(P,a)$-, and $(P,e)$-stable polygons $SA$ over a countable left-zero monoid $S$ under the condition that the set $A\setminus SA$ is indiscernible over a right-zero monoid.