A prime decomposition theorem for string links in a thickened surface

We prove a prime decomposition theorem for string links in a thickened surface. Namely, we prove that any non-braid string link ⊂ Σ × I, where Σ is a compact orientable (not necessarily closed) surface other than S², can be written in the form =₁ # … # ₘ, where ⱼ,j=1,…,m, is prime string link defined up to braid-equivalence, and the decomposition is unique up to permuting the order of factors in its right-hand side.