It appears to me that there is a substantial amount of combinatorial algebra and geometry supporting the theory of central hyperplane arrangements (See Topics in Hyperplane Arrangements, Aguiar and Mahajan).
What breaks when extending this theory to the affine case? Is there a simple relationship between the affine and central case which makes all of the results of Tits algebras/incidence algebras/Dynkin idempotents apply?
I ask this as a machine learning theorist very interested in leveraging many of the algebraic and combinatoric results of this beautiful field, but in my domain we care strictly about the affine case. Since it appears many of the texts and papers in this area concern central arrangements, I'd like to know if those results somehow apply trivially to the affine case, before I embark on developing a deep understanding of the field.
(I'll also note that there is some relationship between the Tits semigroup of an affine arrangement and subsemigroup of the cone of the arrangment, but is this enough to use a large majority of the results in the field? eg. JKS isomorphism etc. My knowledge of the implications of the foregoing semigroup relationship is too limited to make a judgement on if I should precede.)