  The inclusive breakup of three-fragment projectiles is discussed within a
four-body spectator model. Both the elastic breakup and the non-elastic breakup
are obtained in a unified framework. Originally developed in the 80's for
two-fragment projectiles such as the deuteron, in this paper the theory is
successfully generalized to three-fragment projectiles. The expression obtained
for the inclusive cross section allows the extraction of the incomplete fusion
cross section, and accordingly generalizes the surrogate method to cases such
as (t,p) and (t,n) reactions. It is found that two-fragment correlations inside
the projectile affect in a conspicuous way the elastic breakup cross section.
The inclusive non-elastic breakup cross section is calculated and is found to
contain the contribution of a three-body absorption term that is also strongly
influenced by the two-fragment correlations. This latter cross section contains
the so-called incomplete fusion where more than one compound nuclei are formed.
Our theory describes both stable weakly bound three-fragment projectiles and
unstable ones such as the Borromean nuclei.
