Algebraic geometry over algebraic structures III: Equationally Noetherian property and compactness

TL;DR

This paper explores various generalized forms of the equationally Noetherian property in universal algebraic geometry, introducing new classes of algebras and analyzing their interrelations.

Contribution

It introduces and examines weakly equationally Noetherian, qw-compact, uw-compact, and weakly uw-compact algebras, expanding understanding of algebraic properties in this field.

Findings

Defined new algebraic classes and their properties

Analyzed relationships between five classes of algebras

Provided insights into algebraic compactness and Noetherian conditions

Abstract

In this paper we discuss some special generalizations of equationally Noetherian property which naturally arise in the universal algebraic geometry. We introduce weakly equationally Noetherian, qw-compact, uw-compact, and weakly uw-compact algebras and then examine properties of such algebras. Also we consider the connections between five classes: the class of equationally Noetherian algebras, the class of weakly equationally Noetherian algebras, the class of uw-compact algebras, the class of weakly uw-compact algebras, and the class of qw-compact algebras.