  In this paper which is the completion of [1], we construct the
$A_0(q)$-algebra of $Q$-meromorphic functions on the quantum plane. This is the
largest non-commutative, associative, $A_0(q)$-algebra of functions constructed
on the quantum plane. We also define the notion of quantum subsets of R$^2$
which is a generalization of the notion of quantum disc and charactrize some of
their properties. In the end we study the $Q$-homomorphisms of the quantum
plane.
