Möbius conjugation and convolution formulae

Let P   be a locally finite poset with the interval space Int(P)Int(P), and R   be a ring with identity. We shall introduce the Möbius conjugation μ⁎μ⁎ sending each function f:P→Rf:P→R to an incidence function μ⁎(f):Int(P)→Rμ⁎(f):Int(P)→R such that μ⁎(fg)=μ⁎(f)⁎μ⁎(g)μ⁎(fg)=μ⁎(f)⁎μ⁎(g). Taking P   to be the intersection poset of a hyperplane arrangement AA, we shall obtain a convolution identity for the number r(A)r(A) of regions and the number b(A)b(A) of relatively bounded regions, and a reciprocity theorem of the characteristic polynomial χ(A,t)χ(A,t) which gives a combinatorial interpretation of the values |χ(A,−q)||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,gf,g.