Complementation, the inverse of the reduced product operation, is a technique for systematically
finding minimal decompositions of abstract domains. Filé and Ranzato advanced
the state of the art by introducing a simple method for computing a complement. As an
application, they considered the extraction by complementation of the pair-sharing domain
PS from the Jacobs and Langen's set-sharing domain SH. However, since the result of this
operation was still SH, they concluded that PS was too abstract for this. Here, we show that
the source of this result lies not with PS but with SH and, more precisely, with the redundant
information contained in SH with respect to ground-dependencies and pair-sharing. In fact,
a proper decomposition is obtained if the non-redundant version of SH, PSD, is substituted
for SH. To establish the results for PSD, we define a general schema for subdomains of SH
that includes PSD and Def as special cases. This sheds new light on the structure of PSD
and exposes a natural though unexpected connection between Def and PSD. Moreover, we
substantiate the claim that complementation alone is not sufficient to obtain truly minimal
decompositions of domains. The right solution to this problem is to first remove redundancies
by computing the quotient of the domain with respect to the observable behavior, and only
then decompose it by complementation.