Representability of Lyndon–Maddux relation algebras

Abstract

In 2016, Alm–Hirsch–Maddux defined relation algebras $\mathfrak {L}(q,n)$ 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 $q>2304n^2+1$, then $\mathfrak {L}(q,n)$ is representable, and if $q<2n$, then $\mathfrak {L}(q,n)$ is not representable. In the present paper, we reduced this gap by proving that if $q\ge n(\log n)^{1+\varepsilon }$, then $\mathfrak {L}(q,n)$ is representable.