Free algebras of discriminator varieties generated by finite algebras are atomic

TL;DR

This paper proves that in certain algebraic structures called discriminator varieties generated by finite algebras, all definable pre-orders are atomic, revealing a fundamental property of these algebraic systems.

Contribution

It establishes that all definable pre-orders are atomic in finitely generated free algebras of specific discriminator varieties generated by finite algebras.

Findings

All definable pre-orders are atomic in the specified algebraic setting.

The result applies to finitely generated free algebras of discriminator varieties with finite similarity type.

The proof relies on properties of finite members within these algebraic structures.

Abstract

We prove that all definable pre-orders are atomic, in a finitely generated free algebra of a discriminator variety of finite similarity type which is generated by its finite members.