Show that there are exactly two Monoids with two elements.

Question

I am new to learning monoid and I know basic ideas about monoid like the operation been defined has to be associative and has unity in a set. But as of this question, I do not even know where to start from. Please, I need a clear explanation or justification of why the statement is true.

Answer 1

Hint: Fill out Cayley tables for a set of two elements.

The two monoids are given by $$\begin{array}{c|cc} \times & x & y \\ \hline x & x & y \\ y & y & x\end{array}$$ and $$\begin{array}{c|cc} \times & a & b \\ \hline a & a & a \\ b & a & b\end{array}.$$

Answer 2

Hint: it is clear that one of the two elements has to be the identity $e$. Now for the other element $a$, it is either a unit or not.

Each of these choices determines the Monoid completely. Can you explicitly give the two?