Bergman's right-not-left primitive ring

Let $W$ be the set of surjective analytic functions from $\mathbb R\to \mathbb R$ such that $f$ has positive derivative and $f(x+1)=f(x)+1$ for all $x$.

Let $F=\{\sum_{i=1}^n (a_k\cos(2\pi kx)+b_k\sin(2\pi kx))\mid a_i, b_i\in \mathbb Q, n\in\mathbb N\}$.

Let $G$ be a subgroup of $W$ generated by elements of the form $x+g(x)$ with $g(x)\in F$.

There exists a $p\in\mathbb R$ such that no two distinct elements of $G$ have the same value at $p$.

For $f,g\in W$, we write $f\geq g$ if $f(p)\geq g(p)$.

Let $S=\{g\in G\mid g\geq 1\}$. Let $K$ be a field, and $R=K[S]$ be the semigroup algebra. $R$ is the desired ring.