The Möbius function of a lattice

Abstract The structure of an increasing function on an ordered set induces a recursion on the values of its Mobius function μ. When the increasing function is a translation x→x v y on a lattice with zeta function ξ, the recursion takes the form μ(0, 1)=Σ Σ μ(0, y) ξ(y, z) μ(z, 1), a double summation over all pairs (y, z) of complements of x in L.