Domains closed under exponentiation

Question

Apart from $\mathbb{N}$ and $\mathbb{C}$, which other domains satisfy $\forall x, y \in D, x^y \in D$ ,i.e. are closed under exponentiation?

Answer 1

The set of positive (or nonnegative) even numbers and the set of positive odd numbers.

Answer 2

An exponential ring (E-ring) is a ring R with an exponential operation E, i.e. a homomorphism from the additive group of R into its unit group, i.e. $\rm\ E(x+y)\ =\ E(x)\ E(y)\ $ and $\rm\ E(0)\ =\ 1\:.\:$ Obvious examples are $\rm\ (\mathbb C,\ {\it e}^x)\ $ and $\rm\ (\mathbb R,\ a^x),\ a>0\:.\:$ Usually one excludes the trivial exponential $\rm\ E(x)\ = 1\:,\:$ which is the only possibility in characteristic $\rm\:p\:$ since

$$\rm (E(x)-1)^p\ =\ E(x)^p - 1\ =\ E(p\:x)-1\ =\ E(0)-1\ =\ 0\ \ \Rightarrow\ \ E(x) = 1 $$

Such rings and fields are much-studied by model theorists, e.g. in investigations of generalizations of Tarski's problem on the decidability of the reals with exponentiation, Schanuel's conjecture, etc. Searching on these terms should yield a good entry point into related literature.