Many-generator free self-distributive algebras of embeddings

Abstract: An "LD-algebra" or "shelf" is an algebraic structure with a binary operation * satisfying the left-self-distributive law a*(b*c)=(a*b)*(a*c).  Natural examples include the conjugation operation on any group, and weighted means.  In the 1990s, Laver showed that from one of the strongest set theoretic axioms not known to be inconsistent, "I3", one obtains a free singly-generated LD-algebra of embeddings with a natural "application" operation.  In particular, the surrounding set theoretic structure allows one to prove various purely algebraic statements about singly-generated LD-algebras, some of which to this day are only known under the assumption of I3.  It is thus natural to ask whether such free embedding algebras can be obtained with more generators.  I will talk about recent work with Scott Cramer and Sheila Miller Edwards, also using ideas from Gabe Goldberg, showing that one can indeed get a two generator free LD-algebra from the assumption of I3, and about further work with Sheila Miller Edwards extending this to any number of generators up to continuum.

Mathematics

Audience: researchers in the topic

ANTLR seminar

Series comments: This is the Algebra, Number Theory, Logic and Representation theory seminar.