Which monoids have a faithful irreducible representation?

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: $$M(gf)=\{fgf,gf\}, M(g^2)=\{g^2,fg^2\}\quad\text{(minimal)}$$ $$M(f),M(g) \quad\text{(not minimal)}$$

• Yes, it is known I will answer this later in the day, but it is better irreducible is usually used for matrix reps (and we know what these are too). I prefer the term transitive or minimal. Also the answer depends on whether you allow only total mappings and partial mappings. More to come. Sep 25 at 23:41
• Basically it has one if and only if it has a faithful left action on some (=any) of its minimal left ideals. Rhodes calls these left mapping semigroups with respect to their minimal ideal (he also allows partial mappings which allows other L-classes. There can be represented faithfully then as column monomial matrices over the maximal subgroup of their minimal ideal. Sep 25 at 23:43
• I should say the answer is well known in the finite case. The infinite case I'd have to think a bit. Sep 26 at 0:04
• For (possibly infinite) abelian monoids $M$, if I'm correct, the answer is that $M$ admits a faithful minimal $M$-set iff $M$ is a group.
– YCor
Sep 26 at 14:55
• @Ycor that is correct and proved in the Tully paper in my answer. A commutative semigroup has a faithful transitive representation iff it is a group. Sep 26 at 17:29

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.

• I gave a cleaner rewrite of the argument to drop extraneous stuff. Sep 26 at 2:17
• I added an example for "If M is a finite monoid, then all minimal left ideals of M are isomorphic as left M-sets" now :) Sep 26 at 3:57
• For group actions on topological spaces, "transitive" is much weaker than "minimal".
– YCor
Sep 26 at 7:44
• @YCor For actions on non discrete spaces also these two concepts diverge Sep 26 at 12:05
• Of semigroups I should have said Sep 26 at 12:12