Association schemes on triples and a ternary algebra

Abstract

A generalization of the classical association schemes to a higher dimension is developed which is called an association scheme on triples, in short, an AST, and a corresponding ternary, non-associative algebra is developed generalizing the Bose-Mesner algebra. A large number of examples of ASTs are constructed using block designs, permutation groups, and two-graphs. Several identities in the ternary algebra are obtained. It is shown that, given an AST, one can obtain a 2-design and also a family of (classical) association schemes. The concept of a 2-design which is partially balanced with respect to 3-subsets, is introduced.