If a submonoid is in the union of two submonoids, is it in one of them?

Question

It is well-known that if a subgroup of a group is contained in the union of two subgroups, then it is contained in one of them. I would like to know if the same fact is true for submonoids.

So, if $M$ is a monoid, $S$, $R$ and $T$ submonoids of $M$ such that $S\subseteq R\cup T$, then is it true that either $S\subseteq R$ or $S\subseteq T$?

The proof I know for the case of subgroup uses that if $a$ and $a\cdot b\in H$, $H$ a subgroup, then $b\in H$. But this is not true for submonoids. For example, $2$ and $3=2+1$ are contained in the submonoid $S$ generated by $2$ and $3$ in $\mathbb{N}:=\mathbb{Z}_{\ge 0}$, but clearly $1$ is not in $S$.

Answer 1

The answer is no, and I came up with a counter-example

Take $M=\mathbb{N}$, $S:=\langle 2, 3\rangle$, $R:=\langle 2, 5\rangle$ and $T=\langle 3, 4\rangle$. Then $S=R\cup T$ clearly, but $2\in R$, $2\notin T$, $3\in T$ and $2\notin R$.

By the way, one can show that $S=\mathbb{N}\setminus \{1\}$, $R=\mathbb{N}\setminus \{1,3\}$ and $T=\mathbb{N}\setminus \{1,2,5\}$.

Answer 2

Take the monoid $\{1,x,y\}$ with two elements $x,y$ and neutral element $1$, where $ab=a$ for all $a,b \neq 1$. The submonoids $\{1,x\}$ and $\{1,y\}$ provide a counterexample.

Answer 3

Let $M=\mathbb{Z}$ be the integers under addition. Then, $M$ is the union of the proper submonoids $\mathbb{Z}_{\ge 0}$ (nonnegative integers) and $\mathbb{Z}_{\le 0}$ (nonpositive integers). So, $S=\mathbb{Z}$, $R=\mathbb{Z}_{\ge 0}$, and $T=\mathbb{Z}_{\le 0}$ give a counterexample.