Decomposition of Exponential Distributions on Positive Semigroups

Let (S,·) be a positive semigroup and T a sub-semigroup of S. In many natural cases, an element $$x\in S$$ can be factored uniquely as x=yz, where $$y \in T$$ and where z is in an associated “quotient space” S/T. If X has an exponential distribution on S, we show that Y and Z are independent and that Y has an exponential distribution on T. We prove a converse when the sub-semigroup is $$S_t =\{t^n : n \in\mathbb{N}\}$$ for $$t\in S$$ . Specifically, we show that if Y t and Z t are independent and Y t has an exponential distribution on S t for each $$t\in S$$ , then X has an exponential distribution on S. When applied to ([0,∞), +) and $$(\mathbb{N}, +)$$ , these results unify and extend known results on the quotient and remainder when X is divided by t.