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

Given X ? Z N , X is called a cyclic basis if ( X + X ) ? X = Z N , symmetric if x ? X implies - x ? X , and sum-free if ( X + X ) ? X = ? . We ask, for which m , N ? Z + can the set of non-identity elements of Z N 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 .