  This paper continues the study of Fourier transforms on finite inverse
semigroups, with a focus on Fourier inversion theorems and FFTs for new classes
of inverse semigroups. We begin by introducing four inverse semigroup
generalizations of the Fourier inversion theorem for finite groups. Next, we
describe a general approach to the construction of fast inverse Fourier
transforms for finite inverse semigroups complementary to an approach to FFTs
given in previous work. Finally, we give fast inverse Fourier transforms for
the symmetric inverse monoid and its wreath product by arbitrary finite groups,
as well as fast Fourier and inverse Fourier transforms for the planar rook
monoid, the partial cyclic shift monoid, and the partial rotation monoid.

Group Theory;Rings and Algebras