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.