Bi-interpretability of some monoids with the arithmetic and applications

  • Olga KharlampovichEmail author
  • Laura López
Research Article


We will prove bi-interpretability of the arithmetic \({\mathbb {N}}= \langle N, +,\cdot , 0, 1\rangle \) and the weak second order theory of \({\mathbb {N}}\) 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 \(\Sigma _n\).


Free monoid Partially commutative monoid First order properties 



The first author acknowledges the support of the PSC-CUNY award, jointly funded by the Professional Staff Congress and The City University of New York and a grant from the Simons Foundation.


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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Hunter CollegeCUNYNew YorkUSA
  2. 2.Graduate CenterCUNYNew YorkUSA

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