Let \({\mathcal {A}}\) be a hyperplane arrangement in the d-dimensional vector space \({\mathbb {F}}^d\) . We study one-element extensions \({\mathcal {A}}+H_{\varvec{\alpha },a}\) of \({\mathcal {A}}\) for all \((\varvec{\alpha },a)\in {\mathbb {F}}^{d+1}\) . Their intersection semi-lattices \(L({\mathcal {A}}+H_{\varvec{\alpha },a})\) and other combinatorial invariants, including Whitney polynomials, characteristic polynomials, Whitney numbers and face numbers, can be classified by the intersection lattice of the induced adjoint arrangement of \({\mathcal {A}}\) . As a byproduct, we further establish order-preserving relations on these combinatorial invariants and obtain a decomposition formula for the characteristic polynomials \(\chi ({\mathcal {A}}+H_{\varvec{\alpha },a},t)\) .
