- Date
-
Wednesday 18 December 2024, 11.30 - 12.30
- Abstract
- Self-distributivity arises not just in algebra and topology, but also set theory. Indeed, from one of the strongest axioms from set theory known as I3, which is studied as a possible extension to the standard (ZFC) axioms for the universe of sets, one obtains an algebra E_\lambda of embeddings from some initial segment of the universe of sets to itself, with an "application" operation * satisfying j*(k*l)=(j*k)*(j*l). Laver showed that the subalgebra generated by a single such embedding j under * will in fact be the free shelf on 1 generator; and from the set-theoretic structure associated with this concrete (once you assume I3) example of the free shelf, one obtains theorems about shelves - even finite shelves - that are only known under the assumption of I3. It is thus natural to wonder whether free shelves with a higher number of generators can also be found within E_\lambda. In joint work with Scott Cramer and Sheila Miller, we show that the answer is positive if one assumes a little more: assuming the stronger axiom known as I2, we exhibit a free 2-generator LD-algebra of embeddings. My aim is to present this result in a way accessible to those with no background in set theory.
