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