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An axiomatization for the universal theory of the Heisenberg group

Anthony M. Gaglione, Dennis Spellman

Abstract

The Heisenberg group, here denoted $H$, is the group of all $3\times 3$ upper unitriangular matrices with entries in the ring $\mathbb{Z}$ of integers. A.G. Myasnikov posed the question of whether or not the universal theory of $H$, in the language of $H$, is axiomatized, when the models are restricted to $H$-groups, by the quasi-identities true in $H$ together with the assertion that the centralizers of noncentral elements be abelian. Based on earlier published partial results we here give a complete proof of a slightly stronger result.

An axiomatization for the universal theory of the Heisenberg group

Abstract

The Heisenberg group, here denoted , is the group of all upper unitriangular matrices with entries in the ring of integers. A.G. Myasnikov posed the question of whether or not the universal theory of , in the language of , is axiomatized, when the models are restricted to -groups, by the quasi-identities true in together with the assertion that the centralizers of noncentral elements be abelian. Based on earlier published partial results we here give a complete proof of a slightly stronger result.
Paper Structure (5 sections, 6 theorems, 32 equations)

This paper contains 5 sections, 6 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Pedantic Preliminaries
  3. The Lame Property and the Universal Theory of $H$
  4. The Theory in the Base Language
  5. Questions

Key Result

Lemma 3.2

Given a model $G$ of $\mathcal{Q}(H)\cup \text{diag}(H)$ and a representation $G\leq _{H}UT_{3}(R)$ where $R$ locally residually-$\mathbb{Z}$. The representation satisfies the Lame Property if and only if it satisfies the conjunction of the following two conditions:

Theorems & Definitions (10)

  • Definition 3.1
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • Remark 3.4
  • Proposition 3.5
  • Theorem 3.6
  • Corollary 3.7
  • Corollary 3.8
  • proof : Proof of Theorem \ref{['thm1']}