Undecidability of the elementary theory of Young--Fibonacci lattice

TL;DR

This paper proves that the elementary first-order theory of the Young--Fibonacci lattice is undecidable by encoding arithmetic, extending known results from the Young lattice to this related structure.

Contribution

It establishes the undecidability of the elementary theory of the Young--Fibonacci lattice and demonstrates its maximal definability property, similar to the Young lattice.

Findings

Proves undecidability of the Young--Fibonacci lattice theory

Defines arithmetic within the lattice theory

Extends results from Young lattice to Young--Fibonacci lattice

Abstract

For a poset $(P,\leqslant)$ we consider the first-order theory, that is defined by set $P$ and relation $\leqslant$. The problem of undecidability of combinatorial theories attracts significant attention. Recently A. Wires proved the undecidability of the elementary theory of Young lattice and also established the maximal definability property of this theory. The purpose of this article is to obtain the same results for another graded lattice, which has much in common with Young lattice: Young--Fibonacci lattice. As Wires does for Young lattice, for the proof of undecidability we define Arithmetic into this theory.