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

Given X⊆ZNX⊆ZN, XX is called a cyclic basis   if (X+X)∪X=ZN(X+X)∪X=ZN, symmetric   if x∈Xx∈X implies −x∈X−x∈X, and sum-free   if (X+X)∩X=∅(X+X)∩X=∅. We ask, for which mm, N∈Z+N∈Z+ can the set of non-identity elements of ZNZN be partitioned into mm 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≤72≤m≤7. In this manuscript, we provide constructions of Ramsey algebras for every positive integer mm with 2≤m≤4002≤m≤400, with the exception of m=8m=8 and m=13m=13.