Interpreting a field in its Heisenberg group

TL;DR

This paper demonstrates that a field can be interpreted within its Heisenberg group using parameter-free computable formulas, improving upon a 1960 result by Maltsev and providing explicit definitions.

Contribution

It generalizes Maltsev's result by eliminating parameters and providing explicit formulas for interpreting a field in its Heisenberg group.

Findings

Field $F$ is interpretable in $H(F)$ with parameter-free $\Sigma_1$ formulas.

Two proofs are provided: an existence proof and a direct explicit construction.

Conditions for eliminating parameters from interpretations are established.

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.