On the number of universal algebraic geometries

Abstract

The algebraic geometry of a universal algebra \({\textbf{A}}\) is defined as the collection of solution sets of systems of term equations. Two algebras \({\textbf{A}}_1\) and \({\textbf{A}}_2\) are called algebraically equivalent if they have the same algebraic geometry. We prove that on a finite set A with \(|A|\) there are countably many algebraically inequivalent Mal’cev algebras and that on a finite set A with \(|A|\) there are continuously many algebraically inequivalent algebras.