On the cyclic inverse monoid on a finite set
Let \(\Omega_n\) be a set with \(n\) elements and let \(\mathcal{CI}_n\) be the inverse submonoid of the symmetric inverse monoid on the set \(\Omega_n\) consisting of all restrictions of the elements of a cyclic subgroup of order \(n\) acting cyclically on \(\Omega_n\).\N\NIn this paper under review the author studies the monoid. He proves that the monoid \(\mathcal{CI}_n\) has rank \(2\) (for \(n\geqslant 2\)) and it has \(n2^n-n+1\) elements. The author gives presentations of \(\mathcal{CI}_n\) on \(n+1\) generators and \(\frac{1}{2}(n^2+3n+4)\) relations, and on \(2\) generators and \(\frac{1}{2}(n^2-n+6)\) relations. The author studies the remarkable inverse submonoid \({\mathcal{OCI}}_n\) of \(\mathcal{CI}_n\) constituted by all its order-preserving transformations. He shows that the monoid \({\mathcal{OCI}}_n\) has rank \(n\) and it contains \(3\cdot 2^n-2n-1\) elements. Furthermore, the author exhibits the presentations of the monoid \({\mathcal{OCI}}_n\) on \(n+2\) generators and \(\frac{1}{2} (n^2 + 3n + 8)\) relations, and on \(n\) generators and \(\frac{1}{2}(n^2 + 3n)\) relations.
scientific article; zbMATH DE number 7956105
scientific article