Compactness Conditions in Universal Algebraic Geometry

Abstract

In this article, the properties of being equational noetherian, q ω and u ω -compactness, and equational Artinian are studied from the perspective of the Zariski topology. The equational conditions on the relative free algebras of arbitrary varieties are also investigated and their relations to some logic and model theory notions are obtained. Some applications for the case of the universal algebraic geometry over groups are also introduced.

On the Equationally Artinian Groups

“…Equationally Artinian algebras are introduced in [11] and [12]. In this section, we review this notion for the case of G-groups.…”

“…Free groups, Abelian groups, linear groups over Noetherian rings and torsion-free hyperbolic groups are equationally Noetherian. To see interesting properties of this types of groups, the reader can consult [11] and [16]. This kind of groups have very important roles in algebraic geometry of groups.…”

“…There are many equivalent conditions for the property of being equationally Noetherian, for example, it is known that a group A has this property, if and only if, for any natural number n, every descending chain of algebraic sets in A n is finite. According to this equivalent condition, in [11] and [12], the dual property of being equationally Artinian is defined. A group A is equationally Artinian, if and only if, for any natural number n, every ascending chain of algebraic sets in A n is finite.…”

Direct products, varieties, and compactness conditions

“…We can use results of [6] to find such examples. Let us formulate the result of [6] for groups of language L g (below max − n is the following property of a group G: every ascending chain of normal subgroups H ⊳ G becomes stationary). Since any finitely generated metabelian group has the max − n property, all metabelian groups are 1-equationally Noetherian.…”

“…This problem has positive solutions for many varieties. For example, in [6] it was proved that all elements of a group variety V are L g -equationally Noetherian (L g = {·, −1 , 1} is the group language), if and only if the free group F V (X) ∈ V has the max − n property for every finite set X. It follows that the variety of all metabelian groups satisfies Problem 1.…”