Summary: We introduce abstract interpretation frameworks which are variations on the archetypal framework using Galois connections between concrete and abstract semantics, widenings and narrowings and are obtained by relaxation of the original hypotheses. We consider various ways of establishing the correctness of an abstract interpretation depending on how the relation between the concrete and abstract semantics is defined. We insist upon those correspondence allowing for the inducing of the approximate abstract semantics from the concrete one. Furthermore we study various notions of widening and narrowing as a means of obtaining convergence in the iterations used in abstract interpretation.