Example for a claim

Question

If $H$ is a subsemigroup of a semigroup $S$, then it may happen that, for some $a$ and $b$ in $S$, the sets $aH$ and $bH$ don't coincide and, nonetheless, are not disjoint.

Does there exist an example for this claim?

Thanks a lot!

Answer 1

$1+ \mathbb N \neq 2+ \mathbb N$ and $ 1+ \mathbb N \cap 2+ \mathbb N \neq \emptyset$

Answer 2

Take $S=(\mathbb N,\cdot)$ and $H=2\mathbb N$. Now make $a=1$ and $b=2$ (here we are considering the naturals without $0$).

Answer 3

Minimal counterexample. Take $H = S = \{1, 0\}$ with the usual multiplication of integer. Then $0H = \{0\}$ and $1H = S$.