# Representability of Lyndon-Maddux relation algebras

## TL;DR

This paper investigates the conditions under which Lyndon-Maddux relation algebras are representable, narrowing the known gap between non-representability and representability thresholds.

## Contribution

It improves previous bounds by proving that for q at least n times a logarithmic factor, the algebra is representable.

## Key findings

- If q ≥ n(log n)^{1+ε}, the algebra is representable.
- Previous bounds were q > 2304n^2+1 for representability.
- Non-representability holds if q < 2n.

## Abstract

In Alm-Hirsch-Maddux (2016), relation algebras $\mathfrak{L}(q,n)$ were defined that generalize Roger Lyndon's relation algebras from projective lines, so that $\mathfrak{L}(q,0)$ is a Lyndon algebra. In that paper, it was shown that if $q>2304n^2+1$, $\mathfrak{L}(q,n)$ is representable, and if $q<2n$, $\mathfrak{L}(q,n)$ is not representable. In the present paper, we reduced this gap by proving that if $q\geq n(\log n)^{1+\varepsilon}$, $\mathfrak{L}(q,n)$ is representable.