On the Equational Artinian Algebras

Abstract

Equational Artinian algebras were introduced in our previous work: {\em Equational conditions in universal algebraic geometry, to appear in Algebra and Logic, 2015}. In this note, we define the notion of {\em radical topology with respect to an algebra $A$} and using the well-known K\"onig lemma in graph theory, we show that the algebra $A$ is equational Artinian iff this topology is noetherian. This completes the analogy between equational noetherian and equational Artinian algebras.