We present a construction that associates with a given relation *R: M --> A* another relation *R': M --> A(n)* (where *A(n)* is the "*n*-factorization expansion" of *A*). *R'* computes the same cover as *R*, but has some nice properties with respect to pulling back idempotents or (internal) *L* - chains in *A(n)*.
We then exploit this construction to show that for any pseudovariety **A** closed under *A(n)* (e.g. **A** = Aperiodics) the "**A**-idempotent pointlike sets of *M"* are precisely the **A**-pointlike sets of *M* which are idempotents.
The construction may also be used to show that (if *A* is in addition aperiodic)
stabilizers of *A*, pulled back to *M*, are (unions of internal) *L* - chains in *Pl**(***A**,*M)* (the monoid of **A**-pointlike sets of *M*).

Sponsored in part by the FCT approved
projects POCTI 32817/99 and POCTI/MAT/37670/2001 in participation with
the European Community Fund FEDER and by FCT through *Centro de Matemática
da Universidade do Porto.* Also sponsored in part by FCT, the
*Faculdade de Ciências da Universidade do Porto, Programa Operacional Ciência, Tecnologia, Inovação do Quadro Comunitário de Apoio III*, and by *Caixa Geral de Depósitos*