Centralizers of monoids containing the symmetric group

For a monoid M of k-valued unary functions, the centralizer M* of M is the set of k-valued multi-variable functions which commute with every function in M. In this paper, we determine centralizers for all monoids, which contain the symmetric group. For most of such monoids the centralizer turns out to be the least clone. Secondly, we study the monoid M_n of linear unary functions on 2ⁿ, which emerged from the above research, and characterize its centralizer.