Monk Algebras and Representability

TL;DR

This paper investigates the representability of certain relation algebras, disproves a previous generalization, and provides new finite cyclic group representations for specific algebras, advancing understanding of minimal non-representable cases.

Contribution

It refutes a previous proposition, answers an open problem negatively, and introduces the first finite cyclic group representations for several relation algebras.

Findings

Relation algebra 1311_{1316} is not representable.

Proposition 7 from prior work does not generalize.

Finite cyclic group representations are provided for four relation 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 $1314_{1316}$.