Bi-interpretability of Some Monoids with the Arithmetic and Applications

TL;DR

This paper establishes bi-interpretability between the natural numbers with arithmetic and certain monoids, leading to definability results and the QFA property, with implications for the logical complexity of these structures.

Contribution

It proves bi-interpretability of arithmetic with specific monoids, showing definability of substructures and languages, and demonstrates the QFA property for these monoids.

Findings

Finitely generated submonoids are definable.

Recursively enumerable languages are definable.

Monoids have the QFA property.

Abstract

We will prove bi-interpretability of the arithmetic $\N = \langle N, +,\cdot, 0, 1\rangle$ and the weak second order theory of $\N$ with the free monoid $\mathbb{M}_X$ of finite rank greater than 1 and with a non-trivial partially commutative monoid with trivial center. This bi-interpretability implies that finitely generated submonoids of these monoids are definable. Moreover, any recursively enumerable language in the alphabet $X$ is definable in $\mathbb{M}_X$. Primitive elements, and, therefore, free bases are definable in the free monoid. It has the so-called QFA property, namely there is a sentence $\phi$ such that every finitely generated monoid satisfying $\phi$ is isomorphic to $\mathbb{M}_X$. The same is true for a partially commutative monoid without center. We also prove that there is no quantifier elimination in the theory of any structure that is bi-interpretable with $\mathbb N$ to any boolean combination of formulas from $\Pi_n$ or $\Sigma_n$.