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Let $P$ be a locally finite poset with the interval space $\Int(P)$, and $R$a ring with identity. We shall introduce the M\"{o}bius conjugation $\mu^\ast$sending each function $f:P\to R$ to an incidence function$\mu^\ast(f):\Int(P)\to R$ such that $\mu^\ast(fg)=\mu^\ast(f)\ast\mu^\ast(g)$.Taking $P$ to be the intersection poset of a hyperplane arrangement$\mathcal{A}$, we shall obtain a convolution identity for the number$r(\mathcal{A})$ of regions and the number $b(\mathcal{A})$ of relativelybounded regions, and a reciprocity theorem of the characteristic polynomial$\chi(\mathcal{A},t)$, which also leads to a combinatorial interpretation tothe values $|\chi(\mathcal{A},-q)|$ for large primes $q$. Moreover, all knownconvolution identities on Tutte polynomials of matroids will be directconsequences after specializing the poset $P$ and functions $f,g$.

Mobius Conjugation and Convolution Formulae