Remarks

• If you think of C_k,n in the following way, the reason for the name cyclic becomes clear. First there is the beginning piece of the monoid consisting of e, c, c², ..., c^k. Then comes the `cyclic part' consisting of
  c^k, c^{k + 1}, c^{k + 2}, ..., c^{k + n - 1}, c^{k + n} = c^k.

  At the end of this list we are back at the element c^k. After that the cyclic part repeats itself:

  c^{k + n + 1} = c^{k + 1}, c^{k + n + 2} = c^{k + 2}, ...
• |C_k,n| = k + n.
• For every m N with m > 0, there are precisely m nonisomorphic cyclic monoids with m elements, viz., C_{m - k, k} for k = 1, ..., m.
• If k > 0, then no element of C_k,n but e is invertible.
• In C_0,n every element is invertible (in other words, C_0,n is a group, see later).