  Let I be a finitely supported complete m-primary ideal of a regular local
ring (R, m). A theorem of Lipman implies that I has a unique factorization as a
*-product of special *-simple complete ideals with possibly negative exponents
for some of the factors. The existence of negative exponents occurs if the
dimension of R is at least 3 because of the existence of finitely supported
*-simple ideals that are not special. We consider properties of special
*-simple complete ideals such as their Rees valuations and point basis. Let (R,
m) be a d-dimensional equicharacterstic regular local ring with m = (x_1, ...,
x_d)R. We define monomial quadratic transforms of R and consider transforms and
inverse transforms of monomial ideals. For a large class of monomial ideals I
that includes complete inverse transforms, we prove that the minimal number of
generators of I is completely determined by the order of I. We give necessary
and sufficient conditions for the complete inverse transform of a *-product of
monomial ideals to be the *-product of the complete inverse transforms of the
factors. This yields examples of finitely supported *-simple monomial ideals
that are not special. We prove that a finitely supported *-simple monomial
ideal with linearly ordered base points is special *-simple.
