November 14
Andrew Brooke-Taylor,
University of Leeds
A 2 generator free LD-algebra of embeddings
A set with a binary operation * is called an LD-algebra if it satisfies a*(b*c)=(a*b)*(a*c); that is, if left multiplication by any given element a is a homomorphism of the structure. The existence of rank-to-rank embeddings - non-trivial elementary embeddings from some V_lambda to itself - lies at the top of the large cardinal hierarchy (at least, the part compatible with Choice). Given a lambda for which such embeddings exist, there is a natural binary operation of 'application' on these embeddings which clearly gives an LD-algebra; Laver showed that in fact, the algebra generated by a single such embedding under application is actually the free LD-algebra on 1 generator. Many other rank-to-rank embeddings must also exist whenever at least one does, so a natural question is whether Laver's result can be pushed further: does the existence of a rank-to-rank embedding give rise to a free 2-generated LD-algebra of embeddings? I will present a recent result joint with Scott Cramer and Sheila Miller Edwards, showing that from a slightly stronger large cardinal assumption (but one still compatible with Choice, namely, I2) we indeed get a free 2-generated LD-algebra of embeddings.