The Myhill isomorphism theorem does not generalize much

The Myhill isomorphism is a variant of the Cantor-Bernstein theorem. It states that, from two injections that reduces two subsets of $\mathbb{N}$ to each other, there exists a bijection $\mathbb{N}\to \mathbb{N}$ that preserves them. This theorem can be proven constructively. We investigate to which extent the theorem can be extended to other infinite sets other than $\mathbb{N}$. We show that, assuming Markov’s principle, the theorem can be extended to the conatural numbers ${\mathbb{N}}_{\mathrm{\infty }}$ provided that we only require that bicomplemented sets are preserved by the bijection. This restriction is essential. Otherwise, the picture is overall negative: among other things, it is impossible to extend that result to either $2\times {\mathbb{N}}_{\mathrm{\infty }}$, $\mathbb{N}+{\mathbb{N}}_{\mathrm{\infty }}$, $\mathbb{N}\times {\mathbb{N}}_{\mathrm{\infty }}$, ${\mathbb{N}}_{\mathrm{\infty }}^2$, $2^{\mathbb{N}}$ or ${\mathbb{N}}^{\mathbb{N}}$.