The Möbius function on Abelian semigroups

Let X be an Abelian semigroup such that the following conditions hold: (i) if x × y = II (II is the identity element), then x = y = II; (ii) the set {{ x, y }: x × y = a } is finite for any a ∈ X . Let Λ be any field, and let ℰ be the algebra of all Λ-valued functions on X . The convolution of u, υ ∈ ℰ is defined by We set ɛ ( x ) = 1 Λ for all x ∈ X . The Möbius function µ is defined as the solution of the equation ɛ * µ = δ ( δ is the Dirac function). The Möbius function is unique (if it exists at all). Some existence conditions are given. If Λ is replaced by the ring of integers, then µ exists if and only if X does not contain nontrivial idempotents.