  We discuss a microscopic framework for phenomenological boson-fermion models
of nuclear structure based on the U($n/m$) type of superalgebras. The
generalized Dyson mapping of fermion collective superalgebras provides a basis
to do so and to understand how collectivity selects the required preservation
of boson plus fermion number as a good quantum number. We also consider the
difference between dynamical and invariant supersymmetries based on possible
supermultiplets of spectra of neighboring odd and even nuclei. We point out
that different criteria exist for choosing the appropriate single particle
transfer operators in the two cases and discuss a microscopically based method
to construct these operators in the case of dynamical supersymmetry.

Nuclear Theory