  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 then 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.
