Sum-free cyclic multi-bases and constructions of Ramsey algebras

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Given X Zₙ, X is called a cyclic basis if (X+X)∪ X=Zₙ, symmetric if x∈ X implies -x ∈ X, and sum-free if (X+X)∩ X=. We ask, for which m, N+ can the set of non-identity elements of Zₙ be partitioned into m symmetric sum-free cyclic bases? If, in addition, we require that distinct cyclic bases interact in a certain way, we get a proper relation algebra called a Ramsey algebra. Ramsey algebras (which have also been called Monk algebras) have been constructed previously for 2≤ m≤ 7. In this manuscript, we provide constructions of Ramsey algebras for every positive integer m with 2≤ m≤ 400, with the exception of m=8 and m=13.