On association schemes of finite exponent

The notion of an association scheme is a far reaching generalization of the notion of a group. During the past twenty years, many concepts and results in finite group theory have been generalized to finite scheme theory. In the present paper, we suggest two generalizations of the notion of the order of an element of a finite group by defining the girth and the strong girth of a relation of an association scheme on a finite set. We will show that the strong girth of a relation of a scheme is equal to its girth if its strong girth is finite.Similarly to the exponent of a finite group we define the exponent of an association scheme to be the least common multiple of the strong girths of its relations if all of them are finite. We will see that the exponent of a quotient scheme of a scheme S over a normal closed subset of S divides the exponent of S.An element s of an association scheme S will be called regular if s∗ss={s}, and we will call an association scheme regular if each of its elements is regular. We will see that each regular association scheme is of finite exponent and that all association schemes of odd exponent are regular. Furthermore, we will show that any regular association scheme has a non-trivial thin radical. An application to commutative association schemes of finite order concludes our investigation.