# Complexity in Young's Lattice

## TL;DR

This paper explores the logical complexity of Young's lattice, revealing its undecidable elementary theory and the properties of its definable relations, ideals, and conjectured theories.

## Contribution

It characterizes the maximal definability in Young's lattice and establishes its undecidability and non-finite axiomatizability, advancing understanding of its logical structure.

## Key findings

- Young's lattice has an undecidable elementary theory.
- Every ideal generates a finitely axiomatizable universal class.
- Conjectures on the complexities of $\\Sigma_1$ and $\Sigma_2$-theories.

## Abstract

We investigate the complexity of the partial order relation of Young's lattice. The definable relations are characterized by establishing the maximal definability property modulo the single automorphism given by conjugation; consequently, as an ordered set Young's lattice has an undecidable elementary theory and is inherently non-finitely axiomatizable but every ideal generates a finitely axiomatizable universal class of equivalence relations. We end with conjectures concerning the complexities of the $\Sigma_1$ and $\Sigma_2$-theories.