For simply connected nilpotent Lie groups G we show that convolution roots (or—more generally—solutions of mixture-of-(convolution-)power equations) of probability measures with exponentially decreasing tail are uniquely determined. A similar property (up to scalars) holds in the algebra Full-size image (<1 K) of complex- p -adic-valued measures on grids Γ in G . Also, Full-size image (<1 K) has no zero divisors. Furthermore, it is proved that on the semigroups of upper triangular matrices with entries in Full-size image (<1 K) and 1 's on the diagonal, Poisson semigroups {μt}t≥0 are uniquely determined by μ1 .