Minimum number of generator assemblies

Hi everyone. I'm new, and I'm not a native speaker of english, so I hope I won't get any of the advanced algebra names wrong. This is my question:

There is a finite semigroup of functions from the set to with the operation of function composition, with a set of generators . We know that there exists a sequence of assemblies of these generators so that it produces a constant function. What is an upper bound on the length of the minimum number of assemblies for a specified ?