I'm asking this question inspired by the similar question about group and its subgroups. I tried to modify the proof presented there to work for monoids but I failed. I'm also not able to find any counterexample. So my question:
May a monoid have two disjoint submonoids?
or in other words:
May a monoid have a different neutral element than its submonoid?
If YES: Do you have an example of such structure?
If NO: Can you prove it?
On
Consider $\mathbb N_0\times \mathbb N_0$ with componentwise multiplication. Then $\{\,(n,1)\mid n\in\mathbb N_0\,\}$ and $\{\,(n,0)\mid n\in\mathbb N_0\,\}$ are disjoint submonoids (with neutral elements $(1,1)$ and $(1,0)$, respectively).
In fact, simply consider $\{0,1\}$ with multiplication and the submonoids $\{0\}$ and $\{1\}$.
"May a monoid have a different neutral element than its submonoid?"
By the very definition of a submonoid, no.
But of course there are sub-semigroups which happen to be monoids w.r.t. a different neutral element, for example $(\{0\},*,0)$ inside $(\{0,1\},*,1)$.