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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?

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What do you mean by $x^y$ in an arbitrary domain? ($x^y$ is not even completely defined over the complex numbers.) Also, you seem to be using the word "domain" in a nonstandard way; the definition I know is that a domain is a ring with no zero divisors, and in particular it needs to be closed under subtraction, which $\mathbb{N}$ isn't. – Qiaochu Yuan Jan 7 '12 at 9:53
@QiaochuYuan My knowledge of set theory is quite minimal, so I tend to inappropriately generalize terminology. Would 'set' have been more appropriate here? – A Moshbear Jan 7 '12 at 19:19

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.

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

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Actually, I think the set of nonnegative even number does not work because $2^0 = 1$ is odd. In the same vein as your example, for any $n \ge 2$, you have the set of positive multiples of $n$, and the set of positive numbers which can be written $1+kn$ with $k \ge 0$. And more generally any subset of the positive numbers that is stable under product (for example the set of positive squares, the set of positive cubes, the set of number that are coprime to 6,...). – Joel Cohen Jan 8 '12 at 1:46

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