Reflection of elementary embedding axioms on the L[Vλ+1] hierarchy

Say that the property Phi(lambda) of a cardinal lambda strongly implies the property Psi(lambda). If and only if for every lambda, Phi(lambda) implies that Psi(lambda) and that for some lambda' < <lambda>, Psi(lambda'). Frequently in the hierarchy of large cardinal axioms, stronger axioms strongly imply weaker ones. Some strong implications are proved between axioms of the form "there is an elementary embedding j : L alpha [Vlambda +1] --> L-alpha[Vlambda +1] with lambda = sup(n)j(n)(cr(j))". (C) 2001 Elsevier Science B.V. All rights reserved. AMS classification: 03E55.