# Algebra Seminar talk

2022-11-11

Philipp **Lücke***Partition properties and definability*

Abstract:

A Jónsson algebra is an algebra with no proper subalgebras of the same cardinality, i.e., a set $A$ equipped with countably many functions from finite products of $A$ to $A$ that has the property that no proper subset of the same cardinality is closed under all these functions.
It is easy to see that there is a countably infinite Jónsson algebra. Moreover, if there exists a Jónsson algebra of some infinite cardinality $\kappa$, then there exists such an algebra of the next higher cardinality $\kappa^+$.
In contrast, the question whether there is a Jónsson algebra whose cardinality is equal to the first limit cardinal $\aleph_\omega$ is a long-standing open problem in set theory that motivated central developments in this field.

In my talk, I want to present a new approach to restrict the class of models of set theory in which no Jónsson algebras of size $\aleph_\omega$ exist that is based on an analysis of set-theoretic definability at this cardinal.

This is joint work in progress with Omer Ben-Neria (Jerusalem).