Centralizing Monoids, Majority Operations and the Słupecki Clone

A centralizer is a set of operations which commute with some set $F$ of operations and a centralizing monoid $M$ is the unary part of some centralizer. Here $F$ is called a witness of $M$. We give a new proof of the fact that a centralizing monoid having a constant operation as its witness is a maximal centralizing monoid. Then, we present some relations between the binary part of a centralizer having a majority operation as its witness and the Slupecki clone, which is a maximal clone consisting of unary operations and non-surjective operations.