Monk Algebras and Representability
Jeremy F. Alm
TL;DR
This work advances the study of Monk algebras and Ramsey theory by addressing open problems from Kramer and Maddux. It uses Comer’s finite-field construction to analyze color algebras $\mathfrak{E}_n(\{2,3\})$ and their atom-splittings, proving that Prop. 7 cannot be generalized to all $n$ and giving a negative answer to Problem 1.1 by showing $1311_{1316}$ is not representable, while also providing the first finite cyclic-group representations for $31_{37}$, $32_{65}$, $1306_{1314}$, and $1314_{1316}$. The paper contributes concrete finite-field and SAT-based proofs, including a 71-point representation for $1314_{1316}$ and a detailed non-representability argument for $1311_{1316}$, and it outlines open questions regarding weak representability and non-finite-field representations. Overall, it deepens the understanding of representability boundaries in relation algebras and proposes new avenues for identifying the smallest weakly representable but not representable algebras.
Abstract
In ``Monk Algebras and Ramsey Theory,'' \emph{J. Log. Algebr. Methods Program.} (2022), Kramer and Maddux prove various representability results in furtherance of the goal of finding the smallest weakly representable but not representable relation algebra. They also pose many open problems. In the present paper, we address problems and issues raised by Kramer and Maddux. In particular, we prove that their Proposition 7 does not generalize, and we answer Problem 1.1 in the negative: relation algebra $1311_{1316}$ is not representable. Thus $1311_{1316}$ is a good candidate for the smallest weakly representable but not representable relation algebra. Finally, we give the first known finite cyclic group representations for relation algebras $31_{37}$, $32_{65}$, $1306_{1314}$, and $1314_{1316}$.