  We introduce the class of Cartan triples as a generalization of the notion of
a Cartan MASA in a von Neumann algebra. We obtain a one-to-one correspondence
between Cartan triples and certain Clifford extensions of inverse semigroups.
Moreover, there is a spectral theorem describing bimodules in terms of their
support sets in the fundamental inverse semigroup and, as a corollary, an
extension of Aoi's theorem to this setting. This context contains that of
Fulman's generalization of Cartan MASAs and we discuss his generalization in an
appendix.
