Discrimination in a General Algebraic Setting.

Benjamin Fine,Seymour Lipschutz,Anthony Gaglione,Dennis Spellman

doi:10.1155/2015/824268

Benjamin Fine, Seymour Lipschutz + Show 2 more

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https://doi.org/10.1155/2015/824268

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Abstract

Discriminating groups were introduced by G. Baumslag, A. Myasnikov, and V. Remeslennikov as an outgrowth of their theory of algebraic geometry over groups. Algebraic geometry over groups became the main method of attack on the solution of the celebrated Tarski conjectures. In this paper we explore the notion of discrimination in a general universal algebra context. As an application we provide a different proof of a theorem of Malcev on axiomatic classes of Ω-algebras.

Highlights

The notion of a discriminating group was introduced by Baumslag et al in [2] as an outgrowth of the theory of algebraic geometry over groups
A more general class of groups termed squarelike groups was introduced in [3] by Fine et al This class was subsequently shown to be the axiomatic closure of the class of discriminating groups [4]
In an effort to make this paper relatively self contained, we develop in Section 2 universal algebra and in Section 3 we present an overview of the logic and model theory we must apply

Summary

Introduction

The notion of a discriminating group (distinct from an older notion due to Neumann [1]) was introduced by Baumslag et al in [2] as an outgrowth of the theory of algebraic geometry over groups. In [5] Belegradek observed that these notions should be universal, in the sense of universal algebra, and the analogues of the definitions and the proofs of many of the theorems go through in a general algebraic context. In an effort to make this paper relatively self contained, we develop in Section 2 universal algebra and in Section 3 we present an overview of the logic and model theory we must apply.

Universal Algebra

Model Theory and Logic

Discriminating and Squarelike Algebras

On a Theorem of Malcev