The lattice of arithmetic progressions

Marcel K. Goh, Jad Hamdan

Published 2021-06-10Version 1

In this paper we investigate properties of the lattice $L_n$ of subsets of $[n] = \{1,\ldots,n\}$ that are arithmetic progressions, under the inclusion order. For $n\geq 4$, this poset is not graded and thus not semimodular. We start by deriving properties of the number $p_{nk}$ of arithmetic progressions of length $k$ in $[n]$. Next, we look at the set of chains in $L_n' = L_n\setminus\{\emptyset,[n]\}$ and study the order complex $\Delta_n$ of $L_n'$. Third, we determine the set of coatoms in $L_n$ to give a general formula for the value of $\mu_n$ evaluated at an arbitrary interval of $L_n$. In each of these three sections, we give an independent proof 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 conclude by computing the homology groups of $\Delta_n$, providing yet another explanation for the formula of the M\"obius function of $L_n$.