  We introduce two kinds of structures, called v-structures and t-structures,
on biquandles. These structures are used for colorings of diagrams of virtual
links and twisted links such that the numbers of colorings are invariants.
Given a biquandle or a quandle, we give a method of constructing a biquandle
with these structures. Using the numbers of colorings, we show that Bourgoin's
twofoil and non-orientable virtual $m$-foils do not represent virtual links.
