Interpreting a field in its Heisenberg group
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 Montalbán. 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- June 2020
- DOI:
- 10.48550/arXiv.2006.11805
- arXiv:
- arXiv:2006.11805
- Bibcode:
- 2020arXiv200611805A
- Keywords:
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- Mathematics - Logic;
- 03C57 (Primary) 03D45;
- 20H20;
- 12L12
- E-Print:
- Published online by the *Journal of Symbolic Logic*, 23 December 2021. Print version to appear subsequently