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 $t\in S$.



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> Is there a characterization, or a name for such monoids?
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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:
$$M(gf)=\{fgf,gf\}, M(g^2)=\{g^2,fg^2\}\quad\text{(minimal)}$$
$$ M(f),M(g) \quad\text{(not minimal)}$$



Which monoids have a faithful irreducible representation?
**Cleaner rewrite:**


I have a bit more time, so here is a cleaner rewrite. This notion is usually called transitive rather than irreducible, although the terms irreducible and minimal are both used.


If $M$ has a minimal left ideal $L$,then $L$ is a transitive $M$-set and every transitive action is a quotient of $L$ and hence $M$ has a faithful transitive action iff it acts faithfully on $L$. In particular, if $M$ is finite then all its minimal left ideals are isomorphic as $M$-sets (since they are quotients of each other and finite) and $M$ has faithful transitive action iff it acts faithfully on one (equals all) of its minimal left ideals.


The proof is trivial. If $S$ is a transitive $M$-set, then $Ls$ is invariant and hence $Ls=S$ for all $s\in S$. Thus, if we fix $s\in S$, then $m\mapsto ms$ is a surjective $M$-set map $L\to S$.


There is some information on Clifford and Preston Volume 2, Chapter 11.5 on the infinite case. For instance they show noncommutative free monoids have a faithful and transitive action and nontrivial free products of semigroups. See also Tully, E. J. (1961). Representation of a Semigroup by Transformations Acting Transitively on a Set. American Journal of Mathematics, 83(3), 533.


**Further Remark** It is more common to consider transitive actions by partial mappings because many more semigroups have faithful transitive actions in this setting.


