Am I correct that the hyperexponential exp_omega is a bijection on positive infinite surreals?

An exponential level is an equivalence class for the relation $a \asymp_L b \Leftrightarrow \exists n,k \in \mathbb{N}, \exp_n(2^{-k} \log_n(a))\leqslant b \leqslant \exp_n(2^k \log_n(a))$ on the class of positive infinite numbers, where $\exp_n$ and $\log_n$ denote the $n$-fold iterates of $\exp$ and $\log$.

Am I correct that an exponential level is a convex interval?

If so, what is the relationship between them?

Am I correct that the hyperexponential $\exp_{\omega}$ is a bijection on positive infinite surreals?

An exponential level is an equivalence class for the relation $a \asymp_L b \Leftrightarrow \exists n,k \in \mathbb{N}, \exp_n(2^{-k} \log_n(a))\leqslant b \leqslant \exp_n(2^k \log_n(a))$ on the class of positive infinite numbers, where $\exp_n$ and $\log_n$ denote the $n$-fold iterates of $\exp$ and $\log$.

Am I correct that an exponential level is a convex interval?

If so, what is the relationship between them?

Am I correct that the hyperexponential exp_omega is a bijection on positive infinite surreals?

Am I correct that an exponential level (an equivalence class of <>E) is a convex interval?

If so, what is the relationship between them?

Am I correct that the hyperexponential exp_omega is a bijection on positive infinite surreals?

An exponential level is an equivalence class for the relation $a \asymp_L b \Leftrightarrow \exists n,k \in \mathbb{N}, \exp_n(2^{-k} \log_n(a))\leqslant b \leqslant \exp_n(2^k \log_n(a))$ on the class of positive infinite numbers, where $\exp_n$ and $\log_n$ denote the $n$-fold iterates of $\exp$ and $\log$.

Am I correct that an exponential level is a convex interval?

If so, what is the relationship between them?