Varieties defined by linear equations have the amalgamation property

TL;DR

This paper proves that varieties defined by linear equations possess the strong amalgamation property, and under certain conditions, also have the joint embedding property, with implications for classical Maltsev conditions.

Contribution

It establishes that linear-equation-defined varieties have the strong amalgamation property and, with additional conditions, the joint embedding property, extending understanding of their structural features.

Findings

Varieties with linear equations have the strong amalgamation property.

Under specific conditions, these varieties also have the joint embedding property.

Certain properties are preserved even when unary operations are added.

Abstract

A variety is a class of algebraic structures axiomatized by a set of equations. An equation is linear if there is at most one occurrence of an operation symbol on each side. We show that a variety axiomatized by linear equations has the strong amalgamation property. Suppose further that the language has no constant symbol and, for each equation, either one side is operation-free, or exactly the same variables appear on both sides. Then also the joint embedding property holds. Examples include most varieties defining classical Maltsev conditions. In a few special cases, the above properties are preserved when further unary operations appear in the equations.