Varieties generated by completions

TL;DR

This paper investigates the properties of varieties generated by completions of algebras, showing they do not encompass all relation algebras and establishing the existence of many intermediate varieties.

Contribution

It proves that persistently finite algebras are not generated by completions, and demonstrates the existence of continuum many varieties between specific algebraic classes.

Findings

Finite measurable relation algebras are persistently finite.

The variety generated by completions of representable relation algebras does not include all relation algebras.

There are continuum many varieties between certain algebraic classes.

Abstract

We prove that persistently finite algebras are not created by completions of algebras, in any ordered discriminator variety. A persistently finite algebra is one without infinite simple extensions. We prove that finite measurable relation algebras are all persistently finite. An application of these theorems is that the variety generated by the completions of representable relation algebras does not contain all relation algebras. This answers Problem 1.1(1) from a paper by R. Maddux in the negative. At the same time, we confirm the suggestion in that paper that the finite maximal relation algebras constructed in a joint paper by M. Frias and R. Maddux are not in the variety generated by the completions of representable relation algebras. We prove that there are continuum many varieties between the variety generated by the completions of representable relation algebras and the variety of relation algebras.