Elementary Embeddings and Algebra

This chapter describes algebraic consequences of the existence of a non-trivial elementary embedding of V λ into itself (Axiom I3). In addition to composition, the family of all elementary embeddings of the considered structure V λ is equipped with a second binary operation essentially consisting in applying one embedding to another one. This operation satisfies the self-distributivity law x(yz)=(xy)(xz) and, if j is a non-trivial elementary embedding as above, the closure of j under application is a free self-distributive system. Moreover, identifying embeddings that coincide up to some level leads to a sequence of finite quotients, the Laver tables. The main applications obtained so far are a solution to the Word Problem of the self-distributivity law, and a proof that the periods in Laver tables go to infinity with the size. Alternative approaches avoiding elementary embeddings have been found in the first case, but the question of whether a large cardinal hypothesis is necessary remains open in the second.