On when the union of two algebraic sets is algebraic

TL;DR

This paper characterizes when the union of two algebraic sets remains algebraic within certain algebraic structures, providing a comprehensive classification and demonstrating the abundance of such structures of larger sizes.

Contribution

It offers a complete characterization of equational domains in specific algebraic varieties and shows the existence of infinitely many such domains for larger sizes.

Findings

Characterization of equational domains in congruence permutable varieties

Identification of equational domains among small algebras with cyclic automorphisms

Existence of a continuum of equational domains for each size at least three

Abstract

In universal algebraic geometry, an algebra is called an equational domain if the union of two algebraic sets is algebraic. We characterize equational domains, with respect to polynomial equations, inside congruence permutable varieties, and with respect to term equations, among all algebras of size two and all algebras of size three with a cyclic automorphism. Furthermore, for each size at least three, we prove that, modulo term equivalence, there is a continuum of equational domains of that size.