General pairing interactions and pair truncation approximations for fermions in a single- j shell

We investigate Hamiltonians with attractive interactions between pairs of fermions coupled to angular momentum $J$. We show that pairs with spin $J$ are reasonable building blocks for the low-lying states. For systems with only a $J={J}_{max}$ pairing interaction, eigenvalues are found to be approximately integers for a large array of states, in particular, for those with total angular momenta $I\ensuremath{\leqslant}2j$. For $I=0$ eigenstates of four fermions in a single-$j$ shell we show that there is only one nonzero eigenvalue. We address these observations using the nucleon pair approximation of the shell model and relate our results with a number of currently interesting problems.