Consider a closed real interval $[a,b]$ as a toplogical space. Up to homeomeorphism it doesn't matter, but I like to take $[a,b] = [0,\infty]$.

Question 1: What are all of the topological commutative monoid structures $([0,\infty],u,\ast)$ on $[0,\infty]$?

Question 2: What does the space of all topological commutative monoid structures on $[0,\infty]$ look like?

Question 3: What are all of the topological commutative semiring structures on $[0,\infty]$?

Question 4: What does the space of all topological commutative semiring structures on $[0,\infty]$ look like?

Notes:

1. If $([0,\infty],u,\ast)$ is topological (commutative) monoid and $\phi : [0,\infty] \to [0,\infty]$ is a homeomorphism, then $([0,\infty], \phi(u),\phi(\ast(\phi^{-1}(-) \times \phi^{-1}(-)))$ is likewise a topological (commutative) monoid. So it seems reasonable to answer Questions 1 and 2 up to this sort of topological conjugacy.

2. Up to topological conjugacy, we may assume that the unit $u$ is either $0$ or $1$.

3. If the unit is $0$, then $\max$ and $+$ are two topological commutative monoid structures which are not topologically conjugate (since for $a > 0$, $\max(a,-)$, unlike $a+(-)$, is constant on an open set). Are there others?

4. Multiplication $\cdot$ is an interesting topological commutative monoid structures with unit $1$. Are there others?

5. When considering the spaces in (2) and (4), one should not mod out by topological conjugacy. For instance, let $+^\sigma$ be $+$ conjugated by the homeomorphism $x \mapsto x^\sigma$, for some $\sigma \in (0,1]$. Then the function $\sigma \mapsto +^\sigma$ passes through topologically-conjugate commutative monoid structures, but its limit as $\sigma \to 0$ is $\max$, which is in a different topological conjugacy class.

6. I'd be interested in the versions of the questions where commutativity is dropped, though the commutative versions seem like quite enough to think about already.