Complexity in Young's Lattice
Alexander Wires
TL;DR
The paper addresses the definability and logical complexity of the order relation in Young's lattice $\textbf{Y}=\langle \mathcal{P}, \le \rangle$ by introducing a conjugation automorphism and a single constant $[1]+[1]$, establishing a maximal definability property that bi-interprets natural arithmetic. It constructs an explicit arithmetic interpretation using a bijection $\#\colon \mathcal{P} \to \mathbb{N}$ and relations $+_{\#},\times_{\#}$, showing that addition and multiplication are definable by $\Pi_3$-formulas on total partitions, which leads to the undecidability of the $\Sigma_{4}$-theory of $\textbf{Y}^{\ast}$. The main structural result is that $Def(\textbf{Y}^{\ast})=Def(\mathbb{N},+ ,\times)$, i.e., every recursive relation is first-order definable in $\textbf{Y}^{\ast}$, and conversely arithmetic definability can be recovered from partitions. Consequently, the elementary theory of Young's lattice is undecidable and inherently non-finitely axiomatizable, with open questions on the complexities of $\Sigma_1$ and $\Sigma_2$ theories, both with and without constants, and potential extensions to differential posets. This work highlights rich definability phenomena in a classical differential poset and informs the broader study of substructure relations in finite combinatorial structures.
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 $Σ_1$ and $Σ_2$-theories.