  We classify one-element extensions of a hyperplane arrangement by the induced
adjoint arrangement. Based on the classification, several kinds of
combinatorial invariants including Whitney polynomials, characteristic
polynomials, Whitney numbers and face numbers, are constants on those strata
associated with the induced adjoint arrangement, and also order-preserving with
respect to the intersection lattice of the induced adjoint arrangement. As a
byproduct, we obtain a convolution formula on the characteristic polynomials
$\chi(\mathcal{A}+H_{\bm\alpha,a},t)$ when $\mathcal{A}$ is defined over a
finite field $\mathbb{F}_q$ or a rational arrangement.
