On association schemes generated by a relation or an idempotent.

In this paper we study the polynomiality and co-polynomiality of association schemes. These two notions were introduced by Ito. We show that P -polynomial association schemes are co-polynomial and Q -polynomial association schemes are polynomial. We also determine the polynomiality and co-polynomiality for 3-class association schemes. Further we give examples of association schemes that are polynomial but not co-polynomial, or co-polynomial but not polynomial. [ABSTRACT FROM AUTHOR]