On the number of universal algebraic geometries

Abstract

The algebraic geometry of a universal algebra $\mathbf{A}$ is defined as the collection of solution sets of term equations. Two algebras ${\mathbf{A}}_1$ and ${\mathbf{A}}_2$ are called algebraically equivalent if they have the same algebraic geometry. We prove that on a finite set $A$ with $|A|>3$ there are countably many algebraically inequivalent Mal'cev algebras and that on a finite set $A$ with $|A|>2$ there are continuously many algebraically inequivalent algebras.