Representability of relation algebra 1308₁₃₁₆

Determine whether relation algebra 1308₁₃₁₆ is representable.

Background

Relation algebra 1308₁₃₁₆ is obtained by splitting a diversity atom in the color algebra E4({2,3})\mathfrak{E}_4(\{2,3\}) and adding back the mandatory cycle rrr˘rr\breve r. Kramer and Maddux posed its representability as Problem 1.3.

The paper studies the subalgebra 333733_{37} and determines which cyclic groups of order at most 100 support representations of that subalgebra, but it does not resolve representability of the full algebra 1308₁₃₁₆.

References

Problem 1.3 from asks if $1308_{1316}$ is representable. While we have not been able to answer this question, we can provide some information on a subalgebra of $1308_{1316}$.

— Monk Algebras and Representability  (2501.07332 - Alm, 13 Jan 2025) in Section “$1308_{1316}$”