A Ternary Algebra Arising from Association Schemes on Triples Original Research Article

A ternary algebra formed by three-dimensional arrays with entries from an arbitrary field is investigated. The three-dimensional adjacency arrays of a combinatorial structure called an "association scheme on triples" (AST), developed by the authors earlier, form such an algebra, a generalization of the "Bose-Mesner algebra" of the classical association schemes. We define the notion of identity pair and inverse pair and develop methods for finding inverse pairs. For the ternary sub-algebras generated by the adjacency matrices of the nontrivial relations of ASTs constructed from the the groups PSL(2, q), AGL(1, q), and also by 2-designs, it is shown that there is an abundance of inverse pairs.