Examples of proper loops in $\mathbb{R}$

Question

A loop $(L, \cdot)$ is a binary structure that satisfies every group axiom except for the associative property. A loop which is not a group is called a proper loop. A topological loop $(L,\cdot)$ is a topological space which is also a loop such that "$\cdot$" and the inverse operations are continuous.

In the literature there are many examples of finite proper loops, but I couldn't find any example on $\mathbb{R}$. So my question is

What are some examples of proper topological loop structures on $\mathbb{R}$?

The operations here need to be continuous with respect to the usual topology of $\mathbb{R}$. If the loop happens to be commutative, even better.

I am just trying to picture them, because a continuous group structure in $\mathbb{R}$ is, basically, just the addition and there is nothing counterintuitive about it. However, for continuous loops defined on $\mathbb{R}$ there is a very deep theory behind their classification, so examples of them could be very enlightening.

Edit No2: As Eric noticed some of the efforts made towards the classification of continuous loops may provide us with examples. Here is a collection of the most relevant results, which were found in Chapter 18 of the book "Loops in Group and Lie Theory", by Nagy and Strambach.

1. (Thm. 18.18) A topological loop on $\mathbb{R}$ is a proper loop if and only if the group $G$ generated by its translations is not locally compact ($G$ is equipped with the Arens topology).
2. (by Hoffman, pg. 243) A monassociative (a special type of loop) loop on $\mathbb{R}$ is either a Lie group, or the union of two one-parameter semigroups meeting in the unit $e$, each being isomorphic to the semigroup of positive real numbers with respect to the addition.

The first one is impressive, but difficult for me to work with. The second one is very promising to provide an example. So $L=A\cup B$, $A\cap B=\{e\}$ and each component is just the $[0,\infty)$. What boggles me here is that is that we don't seem to know enough to be able to reconstruct the operation "$\cdot$" on $L=A\cup B$ from the available data. For $x, y\in A$, we know what $x\cdot y$ is equal to, it's just $f(x)+f(y)$ where $f:(A, \cdot) \rightarrow [0,\infty)$ is our isomorphism. Similarly, for $x, y\in B$. But for pairs $(a,b)$ such that $a\in A$ and $b\in B$, what can we tell for their product $a\cdot b$?

Answer 1

Two remarkable examples of proper loops $(\mathbb{R}, \ast)$ I've recently discovered.

The first one belongs to Helmut Salzmann: $$x\ast t =\begin{cases} x, &\text{ if }\ t = 0, \\ x+\frac{1}{2}t, &\text{ if }\ t\neq 0\ \text{and}\ \frac{x}{t}\in (-\infty, -\frac{3}{2}]\cup [1,+\infty), \\ \frac{1}{2}x+t, &\text{ if }\ t\neq 0\ \text{and}\ \frac{x}{t}\in [-\frac{2}{3}, 1] , \\ 2x+2t,&\text{ if }\ t\neq 0\ \text{and}\ \frac{x}{t}\in [-\frac{3}{2}, -\frac{2}{3}]. \end{cases}$$ What is really extraordinary about this example is that "$\ast$" behaves a lot like the real addition and yet it fails to give a group operation: It is commutative, its unit is zero and the inverse of each $x$ is $-x$. It also shares some other group properties with the real addition, for example $(x\ast y)^{-1}=y^{-1}\ast x^{-1}$, for every $x, y$.

The second one belongs to Peter Nagy: $$ \DeclareMathOperator{\arccot}{arccot} x\ast y =\begin{cases} x+\arccot (2\nu(x)+ e^{-2\delta(x)}\cot y) +n\pi , &\text{ if } \ 0<y-n\pi<\pi, \\ x+n\pi,&\text{ if } \ y=n\pi, \end{cases}$$ where $n\in \mathbb{Z}$ and $\nu, \delta\in C^r(\mathbb{R})$ are functions satisfying a certain inequality. These loops are not just continuous, but also $C^r$. If I understand correctly, Nagy proceeds to prove something quite impressive, that every differentiable loop belonging in a certain class of well studied loops, is actually isomorphic to a loop of this form.