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

Multiple multi-orbit pairing algebras in shell model and interacting boson models