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Elementary Embeddings and Algebra

Chapter

Abstract

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.

Keywords

Word Problem Braid Group Order Type Elementary Embedding Primitive Recursive Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversité de CaenCaen cedexFrance

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