  The notion of exponential Dowling structures is introduced, generalizing
Stanley's original theory of exponential structures. Enumerative theory is
developed to determine the M\"obius function of exponential Dowling structures,
including a restriction of these structures to elements whose types satisfy a
semigroup condition. Stanley's study of permutations associated with
exponential structures leads to a similar vein of study for exponential Dowling
structures. In particular, for the extended r-divisible partition lattice we
show the M\"obius function is, up to a sign, the number of permutations in the
symmetric group on rn+k elements having descent set {r, 2r, ..., nr}. Using
Wachs' original EL-labeling of the r-divisible partition lattice, the extended
r-divisible partition lattice is shown to be EL-shellable.
