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 \(\Sigma _n\).
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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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Communicated by Mikhail V. Volkov.
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Kharlampovich, O., López, L. Bi-interpretability of some monoids with the arithmetic and applications. Semigroup Forum 99, 126–139 (2019). https://doi.org/10.1007/s00233-019-10021-4
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DOI: https://doi.org/10.1007/s00233-019-10021-4