The lattice of arithmetic progressions

This paper concerns the lattice $L_n$ of subsets of $\{1,\ldots,n\}$ that are arithmetic progressions, under the inclusion order. For $n\geq 4$, this poset is not graded and thus not semimodular. We give three independent proofs of the fact that for $n\geq 2$, $\mu_n(L_n) = \mu(n-1)$, where $\mu_n$ is the M\"obius function of $L_n$ and $\mu$ is the classical (number-theoretic) M\"obius function. We also show that $L_n$ is comodernistic, which implies that $L_n$ is EL-labelable. Comodernism is then used to prove that the order complex $\Delta_n$ of the lattice is either contractible or homotopy equivalent to a sphere.