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Define an exponential ring to be a set $X$ together with a collection of subsets $X_n$ for $n \in \mathbb{Z}$, elements $0_n \in X_n$, $1_n \in X_n$ and maps $+_n: X_n \times X_n \to X_n$ and $\cdot_n: X_n \times X_n \to X_n$ such that:
$(X_n, 0_n, 1_n, +_n, \cdot_n)$ is a commutative ring
$X_{...