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Preprint Number 1803

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1803. Rachael Alvir, Wesley Calvert, Grant Goodman, Valentina Harizanov, Julia Knight, Andrey Morozov, Russell Miller, Alexandra Soskova, and Rose Weisshaar
Interpreting a field in its Heisenberg group

Submission date: 21 June 2020


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 Σ_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.

Mathematics Subject Classification: 03C57 (Primary) 03D45, 20H20, 12L12

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Full text arXiv 2006.11805: pdf, ps.

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