Frobenius vectors, Hilbert series and gluings of affine semigroups

Let $S_1$ and $S_2$ be two affine semigroups, and let $S$ be the gluing of $S_1$ and $S_2$. Several invariants of $S$ are related to those of $S_1$ and $S_2$; we review some of the most important properties preserved under gluings. The aim of this paper is to prove that this is the case for the Frobenius vector and the Hilbert series. Applications to complete intersection affine semigroups are also given.