Words don't come easy: strong affine representations of the polycyclic monoids

Each one-dimensional strong affine representation of the polycyclic monoid ${\mathcal{P}}_n$ is induced by a complete system of residues modulo $n$. We completely characterize these representations in the case when the system of residues is an arithmetic sequence. We accomplish this by introducing a closure operator on the set of primitive words over $\{0,1,\dots ,n-1\}$, and we also describe the lattice of closed sets. This characterization covers all one-dimensional strong affine representations of ${\mathcal{P}}_2$.