On the cyclic inverse monoid on a finite set

In this paper we study the cyclic inverse monoid ${\text{CI}}_n$ on a set ${\mathrm{\Omega }}_n$ with $n$ elements, i.e. the inverse submonoid of the symmetric inverse monoid on ${\mathrm{\Omega }}_n$ consisting of all restrictions of the elements of a cyclic subgroup of order $n$ acting cyclically on ${\mathrm{\Omega }}_n$. We show that ${\text{CI}}_n$ has rank $2$ (for $n⩾2$) and $n{2}^n-n+1$ elements. Moreover, we give presentations of ${\text{CI}}_n$ 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. We also consider the remarkable inverse submonoid ${\text{OCI}}_n$ of ${\text{CI}}_n$ constituted by all its order-preserving transformations. We show that ${\text{OCI}}_n$ has rank $n$ and $3\cdot 2^n-2n-1$ elements. Furthermore, we exhibit presentations of ${\text{OCI}}_n$ 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.