Reference request: a cousin to the log semiring

Question

Let $f$ be strictly increasing on $\mathbb{R}$. Then $x \oplus y := f^{-1}(f(x)+f(y))$ gives rise to a strict symmetric monoidal ($\Rightarrow$ commutative monoid) structure on $(\mathbb{R},\ge)$ with monoidal unit $f^{-1}(0)$. If $f(x) := \exp(x/h)$ for $h > 0$, then $([-\infty,\infty],\oplus,-\infty,+,0)$ is "the" widely used log semiring.

If $f(x) := \sigma(x)|x|^p$ for $p > 0$, then $(\mathbb{R},\oplus,0,\cdot,1)$ is also a semiring.

This semiring does not appear to be widely used, and it is not well-known to me. What is it called, and where can I find references to it in the literature?

NB. I imagine that both of these semirings are morally unified under the aegis of "associated homogeneity" in the sense of Gel'fand and Shilov, i.e., with $\log$ a variant of homogeneous of degree zero.

Answer 1

This is not an answer to the OP, but is too long for a comment.

You may want to check Example (1.23) in J.S. Golan's Semirings and their Applications (Springer, 1999). The first sentence reads as follows:

If $(R, +, \cdot\,)$ is a semiring and $X$ is a set together with a bijective function $\delta \colon X \to R$, then the semiring structure on $R$ induces a semiring structure $(X, \oplus, \odot)$ on $X$ with the operations defined by $x \oplus y = \delta^{-1}(\delta(x) + \delta(y))$ and $x \odot y = \delta^{-1}(\delta(x) \cdot \delta(y))$.

Golan mentions that a special case of this construction is considered by A.A. Mullin in: On the algebraic structure of pre-rings, Notices Amer. Math. Soc. 22 (1975), A703.