Generates $\Bbb{R}$ as a closed subgroup v.s. as a closed subsemigroup

Question

I could not find a definition for the following terms:

$\{r_1,...,r_n\}$ a subset of $\Bbb{R}$ generate $\Bbb{R}$ as a closed subsemigroup - what does it mean?

Similarly, $\{r_1,...r_n\}$ generate $\Bbb{R}$ as a closed subgroup.

Moreover, this two definitions can not imply each other, since they are conditions for different situations (related to something I'm reading).

If one has also a guess- it'd be helpful! Thank you.

Answer 1

We say that a subset $S$ of $\mathbb R$ generates $\mathbb R$ as a closed semigroup if $\mathbb R $ is the only closed subsemigroup of $\mathbb R$ that contains $S$.

We say that a subset $S$ of $\mathbb R$ generates $\mathbb R$ as a closed subgroup if $\mathbb R $ is the only closed subgroup of $\mathbb R$ that contains $S$.

For an explicit example: $[0,1)$ does not generate $\mathbb R$ is a closed subsemigroup as $\mathbb R^+\cup\{0\}$ is a closed subsemigroup of $\mathbb R$ containing $[0,1)$.

On the other hand $[0,1)$ generates $\mathbb R$ as a closed subgroup.