Some model theory of the Heisenberg group: II. The three generator case
Anthony Gaglione, Dennis Spellman · 2020
The Heisenberg group H is the group of all 3 × 3 upper unitriangular matrices with entries in the ring ℤ of integers. We may paraphrase an instance of a question of A. G. Myasnikov as follows: Is the universal theory of H in the language of H, here denoted Th∀(H), axiomatizable by the set diag(H) of atomic and negated atomic sentences true in H together with the set Q(H) of quasi-identities true in H and the single additional axiom NZCT (or noncentral commutative transitivity) asserting that the centralizer of every noncentral element be abelian? In this paper, we characterize the 3-generator models of diag(H) ∪ Q(H) ∪ {NZCT} and show they are all models of Th∀(H).