Varieties generated by completions
H. Andréka, I. Németi
TL;DR
The paper addresses whether the variety generated by completions of representable relation algebras, $Var(RRA^c)$, equals the full variety of relation algebras $RA$. It develops dense-extension techniques in ordered discriminator varieties and proves that completions cannot create persistently finite algebras, while finite measurable relation algebras are persistently finite. These results yield negative answers to Maddux's Problem 1.1(1) and show that there exist continuum-many varieties strictly between $Var(RRA^c)$ and $RA$, clarifying the limitations of completions in generating all relation algebras and revealing a rich intermediate landscape of varieties between $Var(RRA^c)$ and $RA$. The work also links completions to coset-structure representations of algebras, and demonstrates that the finite-measurability criterion precisely captures representability within $Var(RRA^c)$.
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.