Mobius Conjugation and Convolution Formulae

Let $P$ be a locally finite poset with the interval space $\text{Int}(P)$, 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 ${\mu }^{\ast }(f):\text{Int}(P)\to R$ 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 relatively bounded regions, and a reciprocity theorem of the characteristic polynomial $\chi (\mathcal{A},t)$, which also leads to a combinatorial interpretation to the values $\mathrm{\mid }\chi (\mathcal{A},-q)\mathrm{\mid }$ 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 $f,g$.