What are all of the topological (commutative) monoid structures on a closed interval?

Question

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.

EDIT: Thanks to Benjamin Steinberg's reference suggestion below, I'll have to wait for my library to get a copy of that book, but following the citation trail, I've come across a survey article Linearly Ordered Semigroups: Historical Origins and A. H. Clifford's Influence by Hofmann and Lawson (I'm unclear on the precise bibliographical information here). From there I've learned that the commutative monoid structures in Question 1 (at least if $0$ is assumed to be the unit and $\infty$ is assumed to be an absorbing element for the monoid operation) are called "$I$-semigroups". There are 3 main types -- $\max$, $+$, and the operation $x \ast y = \min(1/2,xy)$ on $[1/2,1]$, called a "nil interval" (here the order has been flipped around; I suppose you could flip it back by working with $[0,1]$ with monoid operation $x\ast y = \max(x+y,1)$). More precisely, up to conjugacy, the latter two are the only $I$-semigroups whose only idempotents are the endpoints of the interval. Moreover, every $I$-semigroup can be constructed from these by concatenating them together repeatedly (with $\max$ used to stitch together the concatenated pieces).