# Reference request: a cousin to the log semiring

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.

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 $$x \odot y = \delta^{-1}(\delta(x) \cdot \delta(y))$$.