  Considered is the multiplicative semigroup of ratios of products of principal
minors bounded over all positive definite matrices. A long history of
literature identifies various elements of this semigroup, all of which lie in a
sub-semigroup generated by Hadamard-Fischer inequalities. Via cone-theoretic
techniques and the patterns of nullity among positive semidefinite matrices, a
semigroup containing all bounded ratios is given. This allows the complete
determination of the semigroup of bounded ratios for 4-by-4 positive definite
matrices, whose 46 generators include ratios not implied by Hadamard-Fischer
and ratios not bounded by 1. For n > 4 it is shown that the containment of
semigroups is strict, but a generalization of nullity patterns, of which one
example is given, is conjectured to provide a finite determination of all
bounded ratios.
