Möbius conjugation and convolution formulae

Let P be a locally finite poset with the interval space Int(P ), and R a ring with identity. We shall introduce the Möbius conjugation μ∗ sending each function f : P → R to an incidence function μ∗(f) : Int(P ) → R such that μ∗(fg) = μ∗(f) ∗μ∗(g). Taking P to be the intersection poset of a hyperplane arrangement A, we shall obtain a convolution identity for the number r(A) of regions and the number b(A) of relatively bounded regions, and a reciprocity theorem of the characteristic polynomial χ(A, t), which also leads to a combinatorial interpretation to the values |χ(A,−q)| for large primes q. Moreover, all known convolution identities on Tutte polynomials of matroids will be direct consequences after specializing the poset P and functions f, g.