On the number of universal algebraic geometries

TL;DR

This paper investigates the diversity of algebraic geometries associated with finite universal algebras, revealing a rich landscape of inequivalent structures depending on the algebra's size and properties.

Contribution

It establishes the existence of countably many algebraically inequivalent Mal'cev algebras and continuum many inequivalent algebras on finite sets of specific sizes.

Findings

Countably many algebraically inequivalent Mal'cev algebras on sets with more than 3 elements.

Continuously many algebraically inequivalent algebras on sets with more than 2 elements.

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 $\lvert A \rvert >3$ there are countably many algebraically inequivalent Mal'cev algebras and that on a finite set $A$ with $\lvert A \rvert >2$ there are continuously many algebraically inequivalent algebras.