The inverse semigroup of partial symmetries of a convex polygon
Let P be a convex polygon with set of vertices \(V(P)=\{v_ 1,v_ 2,...,v_ n\}\), where \(v_ i\) is adjacent to \(v_{i+1}\), for \(i=1,...,n-1\), and \(v_ n\) is adjacent to \(v_ 1\). It is considered as a convex region in the plane enclosed by its edges. A polygon A is called a subpolygon of P if \(V(A)=\{v_{i_ 1},...,v_{i_ m}\}\) is a subset of V(P) with \(i_ 1<...<i_ m\), and in A, \(v_{i_ k}\) is adjacent to \(v_{i_{k+1}}\), for \(k=1,...,m-1\), and \(v_{i_ m}\) is adjacent to \(v_{i_ 1}\). Thus A is a convex region in the plane enclosed by the edges of A; it is a ``subregion'' of the region P. If \(m=2\), A is just a line segment; if \(m=1\), A is a point; if \(m=0\), then \(A=\emptyset\), the empty polygon. It is clear that for a given subset \(W=\{v_{j_ 1},...,v_{j_{\ell}}\}\) of V(P) with \(j_ 1<...<j_{\ell}\), there is a unique subpolygon of P having W as the set of its vertices. Let A and B be two subpolygons of P. An isometry from A to B is any bijection from V(A) onto V(B) which is distance preserving under the usual Euclidean metric in the plane. Let S(P) be the set of all isometries between subpolygons of P (including \(\emptyset)\). Then under composition, S(P) is an inverse semigroup called the semigroup of partial symmetries of P. This is a particular case of the partial automorphism semigroup of a mathematical structure M (when \(M=P)\) which has been studied by various authors for many types of structures M (see, for example [\textit{S. M. Goberstein}, On the structure of partial automorphism semigroups, in ``Lattices, Semigroups, and Universal Algebra'', 71-79 (1990)]). The author studies the ideals and congruences of S(P). In particular, she proves that S(P) is a fundamental inverse semigroup and that the only pure congruence on S(P) is the trivial one. For a uniform polygon P, she finds all semilattice and all Clifford congruences on S(P) and establishes a number of other interesting results.
scientific article; zbMATH DE number 4154745
scientific article