  In this paper, we continue the study of weak multiplier Hopf algebras. We
recall the notions of the source and target maps $\varepsilon_s$ and
$\varepsilon_t$, as well as of the source and target algebras. Then we
investigate these objects further. Among other things, we show that the
canonical idempotent $E$ (which is eventually $\Delta(1)$) belongs to the
multiplier algebra $M(B\otimes C)$ where $B=\varepsilon_s(A)$ and
$C=\varepsilon_t(A)$ and that it is a separability idempotent.
  We also consider special cases and examples in this paper. In particular, we
see how for any weak multiplier Hopf algebra, it is possible to make $C\otimes
B$ (with $B$ and $C$ as above) into a new weak multiplier Hopf algebra. In a
sense, it forgets the 'Hopf algebra part' of the original weak multiplier Hopf
algebra and only remembers the source and target algebras. It is in turn
generalized to the case of any pair of non-degenerate algebras $B$ and $C$ with
a separability idempotent $E\in M(B\otimes C)$. We get another example using
this theory associated to any discrete quantum group (a multiplier Hopf algebra
of discrete type with a normalized cointegral). Finally we also consider the
well-known 'quantization' of the groupoid that comes from an action of a group
on a set. All these constructions provide interesting new examples of weak
multiplier Hopf algebras (that are not weak Hopf algebras).
