How many monoids of order three are there?

Question

http://oeis.org/A058129

In the above link we can see the answer is 7. I have tried counting these and don't get 7. I am not sure what I am doing wrong so could someone go through counting these step by step?

Answer 1

Let $M$ be a monoid with three elements. Let $G$ be its group of units (elements which have an inverse) and let $I$ be its minimal ideal. Note that if $|I| = 1$, then $M$ has a zero. Let denote by $C_n$ the cyclic group of order $n$.

1. If $M = G = I$, then $M = C_3$. Otherwise, $G$ and $I$ are disjoint.
2. If $|G| = 2$ and $|I| = 1$ then $M = C_2 \cup \{0\}$.
3. If $|G| = 1$ and $|I| = 1$, then $M = \{1, a, 0\}$ and two cases occur: either $aa = a$ or $aa = 0$.
4. If $|G| = 1$ and $|I| = 2$, then three possibilities occur: $I = C_2$, $I = \{a, b\}$ with $aa = ba = a$ and $bb = ab = b$ or $I = \{a, b\}$ with $aa = ab = a$ and $bb = ba = b$.

Altogether, this gives 1 + 1 + 2 + 3 = 7 possibilities.