论文标题
解释其海森伯格小组的领域
Interpreting a field in its Heisenberg group
论文作者
论文摘要
我们改进并概括了1960年的Maltsev结果。对于字段$ f $，我们用$ h（f）$ the Heisenberg Group表示$ f $。 Maltsev表明，在$ h（f）$中定义的$ f $的副本，使用具有任意非交通对的$（u，v）$作为参数的存在公式。我们证明，使用无参数的计算$σ_1$公式在$ h（f）$中解释了$ f $。我们给出两个证明。首先是证明的存在，取决于哈里森 - 培训，梅尔尼科夫，R。Miller和Montalbán的结果。此证明允许$ f $的元素在没有固定差异的$ h（f）$中代表。第二个证明是直接的，给出了定义解释的明确构成生存公式，其元素为$ f $在$ h（f）$中代表。查看用于到达$ h（f）$中$ f $的无参数解释的内容，我们提供了足以从解释中消除参数的一般条件。
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.
