On the number of universal algebraic geometries

Abstract

The algebraic geometry of a universal algebra $${\textbf{A}}$$ A is defined as the collection of solution sets of systems of term equations. Two algebras $${\textbf{A}}_1$$ A 1 and $${\textbf{A}}_2$$ A 2 are called algebraically equivalent if they have the same algebraic geometry. We prove that on a finite set A with $$|A|$$ | A | there are countably many algebraically inequivalent Mal’cev algebras and that on a finite set A with $$|A|$$ | A | there are continuously many algebraically inequivalent algebras.