Commutator Identities Obtained by the Magnus Algebra

In this paper, two related commutator identities are established through the use of the Magnus Algebra (the algebra of noncommutative formal power series with integral coefficients)

The Persistence of Universal Formulae in Free Algebras

Gilbert Baumslag, B.H. Neumann, Hanna Neumann, and Peter M. Neumann successfully exploited their concept of discrimination to obtain generating groups of product varieties via the wreath product construction. We have discovered this same underlying concept in a somewhat different context. Specifically, let V be a non-trivial variety of algebras. For each cardinal α let Fα(V) be a V-free algebra of rank α. Then for a fixed cardinal r one has the equivalence of the following two statements ..

The Persistence of Universal Formulae in Free Algebras

Gilbert Baumslag, B.H. Neumann, Hanna Neumann, and Peter M. Neumann successfully exploited their concept of discrimination to obtain generating groups of product varieties via the wreath product construction. We have discovered this same underlying concept in a somewhat different context. Specifically, let V be a non-trivial variety of algebras. For each cardinal α let Fα(V) be a V-free algebra of rank α. Then for a fixed cardinal r one has the equivalence of the following two statements ..

An axiomatization for the universal theory of the Heisenberg group

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.Comment: 13 pages. Published in journal of Groups, Complexity, Cryptolog

Orderable groups, elementary theory, and the Kaplansky conjecture

We show that each of the classes of left-orderable groups and orderable groups is a quasivariety with undecidable theory. In the case of orderable groups, we find an explicit set of universal axioms. We then consider the relationship with the Kaplansky group rings conjecture and show that K, the class of groups which satisfy the conjecture, is the model class of a set of universal sentences in the language of group theory. We also give a characterization of when two groups in K or more generally two torsion-free groups are universally equivalent

Reflections on Discriminating Groups

Here we continue the study of discriminating groups as introduced by Baumslag, Myasnikov and Remeslennikov in [7]. First we give examples of finitely generated groups which are discriminating but not trivially discriminating, in the sense that they do not embed their direct squares, and then we show how to generalize these examples. In the opposite direction we show that if F is a non-abelian free group and R is a normal subgroup of F such that F/R is torsion-free, then F/R\u27 is non-discriminating

Groups Whose Universal Theory Is Axiomatizable by Quasi-Identities

Discriminating groups were introduced in [3] with an eye toward applications to the universal theory of various groups. In [6] it was shown that if G is any discriminating group, then the universal theory of G coincides with that of its direct square G x G. In this paper we explore groups G whose universal theory coincides with that of their direct square. These are called square-like groups. We show that the class of square-like groups is first-order axiomatizable and contains the class of discriminating groups as a proper subclass. Further we show that the class of discriminating groups is not first-order axiomatizable