Semigroup algebra of an idempotent semigroup

Question

Let us consider $S=\{1,2\}$ with the operation $xy=\max\{x,y\}$. Then $S$ is a commutative semigroup with unity. Consider its complex algebra $\mathbb C[S]={\rm span}\{e_1, e_2\}$, where $e_i e_j = e_{ij}$. This algebra is unital as $e_1$ is the unit. Then $e_1, e_2$ and $e_1 - e_2$ are idempotents in this algebra so it cannot be isomorphic to the algebra of upper-triangular matirces with constant diagonal. However, there are only 2 (up to isomorphism) complex unital algebras. $\mathbb C[S]$ so it must be isomorphic to $\mathbb{C}\oplus \mathbb{C}$, right?

Answer 1

Looks like

$$ \begin{align*} e_1 &\leftrightarrow \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \\ e_2 &\leftrightarrow \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}. \end{align*} $$