Undecidability of the elementary theory of Young--Fibonacci lattice

For a poset $(P,⩽)$ we consider the first-order theory, that is defined by set $P$ and relation $⩽$. 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.