  In nuclei with valence nucleons are say identical nucleons and say these
nucleons occupy several-$j$ orbits, then it is possible to consider pair
creation operator $S_+$ to be a sum of the single-$j$ shell pair creation
operators $S_+(j)$ with arbitrary phases, $S_+=\sum_j \alpha_j S_+(j);
\alpha_j=\pm 1$. In this situation, it is possible to define multi-orbit or
generalized seniority that corresponds to the quasi-spin $SU(2)$ algebra
generated by $S_+$, $S_-=(S_+)^\dagger$ and $S_0=(\hat{n} -\Omega)/2$
operators; $\hat{n}$ is number operator and $\Omega=[\sum_j (2j+1)]/2$. There
are now multiple pairing quasi-spin $SU(2)$ algebras. Also, the $\alpha_j$'s
and the generators of the corresponding generalized seniority generating
sympletic algebras $Sp(2\Omega)$ in $U(2\Omega) \supset Sp(2\Omega)$ have
one-to-one correspondence. Using these, derived are the special seniority
selection rules for electromagnetic transitions. A particular choice for
$\alpha_j$'s as advocated by Arvieu and Moszkowski (AM) in the past gives
pairing Hamiltonians having maximum correlation with well known effective
interactions. The various results derived for identical fermion systems are
shown to extend to identical boson systems with the bosons occupying
several-$\ell$ orbits as for example in $sd$, $sp$, $sdg$ and $sdpf$ IBM's. The
quasi-spin algebra here is $SU(1,1)$ and the generalized seniority quantum
number is generated by $SO(2\Omega)$ in $U(2\Omega) \supset SO(2\Omega)$. The
different $SO(2\Omega)$ algebras here will be important in the study of quantum
phase transitions and order-chaos transitions in nuclei.
