  We introduce a multiple conjugation biquandle, and show that it is the
universal algebra to define a semi-arc coloring invariant for handlebody-links.
A multiple conjugation biquandle is a generalization of a multiple conjugation
quandle. We extend the notion of $n$-parallel biquandle operations for any
integer $n$, and show that any biquandle gives a multiple conjugation biquandle
with them.
