In this short note, we show that the number of monogenic submonoids of the full transformation monoid of degree $n$ for $n>0$, equals the sum of the number of cyclic subgroups of the symmetric groups on $1$ to $n$ points. We also prove an analogous statement for monogenic subsemigroups of the finite full transformation monoids, as well as monogenic inverse submonoids and subsemigroups of the finite symmetric inverse monoids.

https://www.tandfonline.com/doi/pdf/10.1080/00927872.2023.2214821?needAccess=true&role=button

Counting monogenic monoids and inverse monoids

Some properties of monogenic inverse semigroups are considered. In particular, in a free monogenic inverse semigroup we study the disposition of idempotents, describe the structure of ideals, classify the congruences.

G. G. Dyadchenko

Papers

Structure of monogenic inverse semigroups