Rank Problems for Composite Transformations

Let (X, F) be a pair consisting of a finite set X and a set F of transformations of X, and, let and F(l) denote, respectively, the semigroup generated by F and the part of consisting of the transformations determined by a generator sequence of length no more than a given integer l. We show the following: • The problem whether or not, for a given pair (X, F) and a given integer r, there is an idempotent transformation of rank r in is PSPACE-complete. • For each fixed r≥1, it is decidable in a polynomial time, for a given pair (X, F), whether or not contains an idempotent transformation of rank r, and, if yes then a generator sequence of polynomial length composing to an idempotent transformation of rank r can be obtained in a polynomial time. • For each fixed r≥1, the problem whether or not, for a given (X, F) and l, there is an idempotent transformation of rank r in F(l) is NP-complete. • For each fixed r≥2, to decide, for a given (X, F), whether or not contains a transformation of rank r is NP-hard.