Let $*$ be a binary operation on a set $M$, with an identity element $e\in M$.

A *monoid representation* of $(M,*,e)$ is a map $\delta:M\to (S\to S)$ for some set $S$, such that $\delta(e)=\mathrm{id}_S$, and $\delta(a*b)=\delta(a)\circ\delta(b)$ for all $a,b\in M$. (A representation could also be called an *action*, I suppose?)

$\delta$ is

*faithful*if $\delta$ is injective as a function from $M$ to $S^S$.$\delta$ is

*irreducible*if there is no subset $\emptyset\subsetneq T\subsetneq S$ with $\delta(m)(t)\in T$ for all $t\in T$ and $m\in M$.

Which monoids $(M,*,e)$ have faithful irreducible representations? For example, all groups do have such representations, but the monoid $\{e,a\}$ with $a^2=a\ne e$ does not since we can take $T=\{\delta(a)(t)\}$ for a fixed $t\in S$.

Is there a characterization, or a name for such monoids?

Example: let $M$ be generated by $f,g:\{0,1,2,3\}\to\{0,1,2,3\}$ where $f(0)=1$, $g(0)=2$, $f(1)=g(1)=1$, and $f(2)=3$, $f(3)=2$, $g(2)=3$, $g(3)=3$. The monoid is $$\begin{matrix} & && e &&\\ &&f & &g\\ &f^2 & gf&&fg & g^2\\ & & fgf&&&fg^2\\ \end{matrix}$$ which has the ideals: $$M(gf)=\{fgf,gf\}, M(g^2)=\{g^2,fg^2\}\quad\text{(minimal)}$$ $$ M(f),M(g) \quad\text{(not minimal)}$$

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