The Order of Generalized Hypersubstitutions of Type τ=(2)

The order of hypersubstitutions, all idempotent elements on the monoid of all hypersubstitutions of type τ 2 were studied by K. Denecke and Sh. L. Wismath and all idempotent elements on the monoid of all hypersubstitutions of type τ 2, 2 were studied by Th. Changpas and K. Denecke. We want to study similar problems for the monoid of all generalized hypersubstitutions of type τ 2 . In this paper, we use similar methods to characterize idempotent generalized hypersubstitutions of type τ 2 and determine the order of each generalized hypersubstitution of this type. The main result is that the order is 1,2 or infinite.