  The weak tensor product was introduced by Snevily as a way to construct new
graphs that admit $\alpha$-labelings from a pair of known $\alpha$-graphs. In
this article, we show that this product and the application to
$\alpha$-labelings can be generalized by considering as a second factor of the
product, a family $\Gamma$ of bipartite $(p,q)$-graphs, $p$ and $q$ fixed. The
only additional restriction that we should consider is that for every $F\in
\Gamma$, there exists and $\alpha$-labeling $f_F$ with $f_F(V(F))=L\cup H$,
where $L,H \subset [0,q]$ are the stable sets induced by the characteristic of
$f_F$ and they do not depend on $F$. We also obtain analogous applications to
near $\alpha$-labelings and bigraceful labelings.
