  Schubert proved that, given a composite link $K$ with summands $K_{1}$ and
$K_{2}$, the bridge number of $K$ satisfies the following equation:
$$\beta(K)=\beta(K_{1})+\beta(K_{2})-1.$$ In ``Conway Produts and Links with
Multiple Bridge Surfaces", Scharlemann and Tomova proved that, given links
$K_{1}$ and $K_{2}$, there is a Conway product $K_{1}\times_{c}K_{2}$ such that
$$\beta(K_{1}\times_{c} K_{2}) \leq \beta(K_{1}) + \beta(K_{2}) - 1$$ In this
paper, we define the generalized Conway product $K_{1}\ast_{c}K_{2}$ and prove
the lower bound $\beta(K_{1}\ast_{c}K_{2}) \geq \beta(K_{1})-1$ where $K_{1}$
is the distinguished factor of the generalized product. We go on to show this
lower bound is tight for an infinite class of links with arbitrarily high
bridge number.
