We give a characterization of star-free languages (a well-known subclass of regular languages) in a λ-calculus with support for non-commutative affine types (in the sense of linear logic), via the algebraic characterization of the former using aperiodic monoids. This was the first result in a research program that Cécilia Pradic (Swansea University) and I have carried out since my PhD, on which I will say a few words.