# Bi-interpretability of some monoids with the arithmetic and applications

- 15 Downloads

## Abstract

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 \(\Sigma _n\).

## Keywords

Free monoid Partially commutative monoid First order properties## Notes

### Acknowledgements

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.

## References

- 1.Cooper, S.B.: Computability Theory. Chapman and Hall/CRC, Boca Raton (2004)zbMATHGoogle Scholar
- 2.Kharlampovich, O., Miasnikov, A.: Tarski-type problems for free associative algebras. J. Algebra
**500**, 589–643 (2017)MathSciNetCrossRefzbMATHGoogle Scholar - 3.Kharlampovich, O., Myasnikov, A.: Elementary theory of free non-abelian groups. J. Algebra
**302**(2), 451–552 (2006)MathSciNetCrossRefzbMATHGoogle Scholar - 4.Kharlampovich, O., Myasnikov, A.: Definable sets in a hyperbolic group. Int. J. Algebra Comput.
**23**(1), 91–110 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 5.Matiyasevich, Y.: Diophantine representation of enumerable predicates. Math. Not. Acad. Sci. USSR
**12**(1), 501–504 (1972)MathSciNetzbMATHGoogle Scholar - 6.Nies, A.: Describing groups. Bulletin of Symbolic Logic
**13**(3), 305–339 (2007)MathSciNetCrossRefzbMATHGoogle Scholar - 7.Ould Houcine, A.: Homogeneity and prime models in torsion-free hyperbolic groups. Conflu. Math.
**3**(1), 121–155 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - 8.Perin, C., Pillay, A., Sklinos, R., Tent, K.: On groups and fields interpretable in torsion-free hyperbolic groups. Münster J. Mat.h.
**7**(2), 609–621 (2014)MathSciNetzbMATHGoogle Scholar - 9.Perin, C., Sklinos, R.: Homogeneity in the free group. Duke Math. J.
**161**(13), 2635–2668 (2012)MathSciNetCrossRefzbMATHGoogle Scholar - 10.Quine, W.: Concatenation as a basis for arithmetic. J. Symb. Logic
**11**(4), 105–114 (1946)MathSciNetCrossRefzbMATHGoogle Scholar - 11.Rogers, H.: The Theory of Recursive Functions and Effective Computability, 2nd edn. MIT Press, Cambridge (1987)Google Scholar
- 12.Sela, Z.: Diophantine geometry over groups VI: the elementary theory of a free group. Geom. Funct. Anal.
**16**, 707–730 (2006)MathSciNetzbMATHGoogle Scholar