Non-trivial semisubgroups of the unit interval?

Question

The open unit interval $(0,1)\subset \mathbb{R}$ forms a semigroup under multiplication. What are examples of subsemigroups of this semigroup, which are not intervals of the form $(0,a)$ for $a\in (0,1)$? Have they been studied? (Where) do they naturally appear? Are there some known classification results?

Answer 1

Consider any subset $P$ of the positive prime integers $\{2,3,5,7,11,13,17,\ldots\}$. Next, consider the set of all numbers of the form $p_1^{-n_1} \cdots p_k^{-n_k}$, where $p_1,\ldots,p_k \in P$, and where $n_1,\ldots n_k \ge 1$ are integers. This gives uncountably many distinct examples (given that the set of positive prime integers has uncountably many distinct subsets).

Answer 2

Using minus the logarithm, you are looking at the subadditivesemi-group of $(0,\infty)$. The set ${\bf N}\backslash\{0\}$ is an example of such subadditivesemi-group.

Answer 3

To add to both the other answers, the group $\mathbb{R}$ is isomorphic to $\bigoplus_c \mathbb{Q}$, so has not just uncountably many, but $2^{2^{\aleph_0}}$ many subgroups, by taking the suspace over $\mathbb{Q}$ generated by any subset of the basis for $\mathbb{Q}$ over $\mathbb{R}$. Any intersection of this with $(0,\infty)$ is a subsemigroup of $(0,\infty)$, and they are all different. So there are $2^{2^{\aleph_0}}$ distinct subsemigroups of $(0,\infty)$, hence $2^{2^{\aleph_0}}$ distinct subsemigroups of $(0,1)^\times$.

Another thing to notice is if you take any element $a \in (0,1)$, the set $ \langle a \rangle = \{a^n|n \in \mathbb{N}-\{0\} \}$ is a subsemigroup. This actually applies to any element of any semigroup, so you can almost always find lots of subsemigroups.