March 21
Sheila Miller Edwards,
Arizona State University
How to construct a free, two-generated left distributive algebra of elementary embeddings
The relationship between left distributivity and very large cardinal embeddings was discovered in the 1980s but remains, in many ways, mysterious. In the late 1980s Richard Laver showed that the closure of a single elementary embedding under the application operation generates a free left distributive algebra and demonstrated the linearity of a particular ordering on terms of the free left distributive algebra (given the existence of such embeddings). Patrick Dehornoy later used the braid group on infinitely many generators to show the linearity of that ordering relation within ZFC. (The consistency strength of other related theorems is still unknown). David Larue subsequently extended that work to demonstrate braid group representations of the free left distributive algebra on $n$ generators, for any natural number $n$. Still elusive was an algebra of embeddings isomorphic to a free left distributive algebra on more than one generator. We present an inverse limit construction of such a free, two-generated left distributive algebra of embeddings from a slightly stronger large cardinal assumption than the one used by Laver. (Joint work with Andrew Brooke-Taylor and Scott Cramer.)