Is max-plus semiring isomorphic to $\mathbb{R_{+}}$

Question

Define two new binary operation $\oplus,\otimes$ on $\mathbb{R_{+}}$(positive real number),$$a\oplus b=max(a,b),\\a\otimes b=a+b,$$ It is easy to see $(\mathbb{R_{+}},\oplus,\otimes)$ is a semiring.I want to know if it is isomorphic to $(\mathbb{R_{+}},+,.)$

I tried using the deformation of $log_{q}x$ to construct the isomorphism.But it doesn't work.

Answer 1

Assuming $\mathbb{R} = \mathbb{R}_{\geq 0}$, otherwise we don't even have additive identity. $0$ is an identity for both $\oplus$ and $\otimes$. This implies that $(\mathbb{R}_+,\oplus,\otimes)$ is not isomorphic to $(\mathbb{R}_+,+,.)$ since $1 \neq 0$ in $(\mathbb{R}_+,+,.)$.