FACTORIZABLE SEMIGROUP OF PARTIAL SYMMETRIES OF A REGULAR POLYGON

The semigroup of partial symmetries of a convex polygon P is the inverse semigroup of all isometries of subpolygons of P, under composition. This semigroup is a natural generalization of the group of symmetries of a polygon, as well as a particular instance of an inverse semigroup formed by taking all isomorphisms between substructures of a given mathematical structure. The general properties of the semigroup of partial symmetries of any convex polygon were explored in [5]; in this paper we restrict consideration to regular polygons where, as in the situation with groups, much more structural information can be obtained. We show that every isometry between subpolygons of P can be extended to an isometry of P and use this to factorize these semigroups into a product of a semilattice and a group. If the number of vertices of the polygon is odd, a complete characterization is given in terms of the group of symmetries of P.