The lattice of arithmetic progressions

Abstract

This paper concerns the lattice $L_n$ of subsets of $\{1,\dots ,n\}$ that are arithmetic progressions, under the inclusion order. For $n\ge 4$, this poset is not graded and thus not semimodular. We give three independent proofs of the fact that for $n\ge 2$, ${\mu }_n(L_n)=\mu (n-1)$, where ${\mu }_n$ is the Möbius function of $L_n$ and $\mu$ is the classical (number-theoretic) Möbius 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.

15 pages, 1 figure, 2 tables. Two new sections have been added: we show the lattice is comodernistic and use this to determine its homotopy type

References in corpus (2)

• A cell complex in number theory
• Arithmetic subsequences in a random ordering of an additive set