A TERNARY ALGEBRA ARISING FROM ASSOCIATION SCHEMES ON TRIPLES

A ternary algebra formed by three-dimensional arrays with entries from an arbitrary field is investigated. The three-dimensional adjacency a rrays of a combinatorial structure called an ''association scheme on t riples'' (AST), developed by the authors earlier, form such an algebra , a generalization of the ''Bose-Mesner algebra'' of the classical ass ociation schemes. We define the notion of identity pair and inverse pa ir and develop methods for finding inverse pairs. For the ternary sub- algebras generated by the adjacency matrices of the nontrivial relatio ns of ASTs constructed from the the groups PSL(2, q), AGL(1, q), and a lso by 2-designs, it is shown that there is an abundance of inverse pa irs. (C) 1994 Academic Press, Inc.