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arXiv · 2006.11805

Interpreting a field in its Heisenberg group

Rachael Alvir·Wesley Calvert·Grant Goodman·Valentina Harizanov·Julia Knight·Andrey Morozov·Russell Miller·Alexandra Soskova·Rose Weisshaar
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Abstract

We improve on and generalize a 1960 result of Maltsev. For a field $F$, we denote by $H(F)$ the Heisenberg group with entries in $F$. Maltsev showed that there is a copy of $F$ defined in $H(F)$, using existential formulas with an arbitrary non-commuting pair $(u,v)$ as parameters. We show that $F$ is interpreted in $H(F)$ using computable $\Sigma_1$ formulas with no parameters. We give two proofs. The first is an existence proof, relying on a result of Harrison-Trainor, Melnikov, R. Miller, and Montalb\'an. This proof allows the possibility that the elements of $F$ are represented by tuples in $H(F)$ of no fixed arity. The second proof is direct, giving explicit finitary existential formulas that define the interpretation, with elements of $F$ represented by triples in $H(F)$. Looking at what was used to arrive at this parameter-free interpretation of $F$ in $H(F)$, we give general conditions sufficient to eliminate parameters from interpretations.

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Rachael Alvir, Wesley Calvert, Grant Goodman, Valentina Harizanov, Julia Knight, Andrey Morozov, Russell Miller, Alexandra Soskova, Rose Weisshaar. 2020-06-21. Interpreting a field in its Heisenberg group. https://doi.org/10.1017/jsl.2021.107

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