Monk algebras and representability

Abstract In “Monk Algebras and Ramsey Theory,” 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 the algebra obtained by splitting the atoms of an n atom Monk algebra is representable for $$n=32$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>32</mml:mn> </mml:mrow> </mml:math> and $$n=116$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>116</mml:mn> </mml:mrow> </mml:math> , and hence Proposition 7 in Kramer-Maddux does not generalize. We answer Problem 1(3) in the negative: relation algebra $$1311_{1316}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mn>1311</mml:mn> <mml:mn>1316</mml:mn> </mml:msub> </mml:math> is not representable. Thus $$1311_{1316}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mn>1311</mml:mn> <mml:mn>1316</mml:mn> </mml:msub> </mml:math> is a good candidate for the smallest weakly representable but not representable relation algebra.