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In this paper, we continue the study of weak multiplier Hopf algebras. Werecall the notions of the source and target maps $\varepsilon_s$ and$\varepsilon_t$, as well as of the source and target algebras. Then weinvestigate these objects further. Among other things, we show that thecanonical idempotent $E$ (which is eventually $\Delta(1)$) belongs to themultiplier 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, wesee how for any weak multiplier Hopf algebra, it is possible to make $C\otimesB$ (with $B$ and $C$ as above) into a new weak multiplier Hopf algebra. In asense, it forgets the 'Hopf algebra part' of the original weak multiplier Hopfalgebra and only remembers the source and target algebras. It is in turngeneralized to the case of any pair of non-degenerate algebras $B$ and $C$ witha separability idempotent $E\in M(B\otimes C)$. We get another example usingthis theory associated to any discrete quantum group (a multiplier Hopf algebraof discrete type with a normalized cointegral). Finally we also consider thewell-known 'quantization' of the groupoid that comes from an action of a groupon a set. All these constructions provide interesting new examples of weakmultiplier Hopf algebras (that are not weak Hopf algebras).

Weak Multiplier Hopf Algebras II. The source and target algebras