For a finite semigroup S and pseudovariety V, (Y, T) is a V-stable pair of S iff Y ⊆ S, T ≤ S and for any relational morphism R : S ⇝ V with V ∈ V there exists a v ∈ V such that Y ⊆ R-1(v) and T ≤ R-1( Stab (v)). X ≤ S is stable if it is generated by an [Formula: see text]-chain {ai} with aiaj = ai for j < i. Given a relation R : S ⇝ A ∈ A (where A denotes the pseudovariety of aperiodic semigroups) that computes PlA(S), we construct a new relation R∞ : S ⇝ (A(M))# that computes A-stable pairs. This proves the main result of this paper: (Y, T) is an A-stable pair of S iff T ≤ ∪ X for some stableX ≤ PlA(S) and Y ⊆ Y' for some Y' ∈ PlA(S) with Y'x = Y' for all x ∈ X. As a corollary we get that if V is a local pseudovariety of semigroups, then V * A has decidable membership problem.