A Study on Centralizing Monoids with Majority Operation Witnesses

A centralizing monoid M is a set of unary operations which commute with some set F of operations. Here, F is called a witness of M . On a 3-element set, a centralizing monoid is maximal if and only if it has a constant operation or a majority minimal operation as its witness. In this paper, we take one such majority operation, which corresponds to a maximal centralizing monoid, on a 3-element set and obtain its generalization, called mb , on a k-element set for any k >= 3. We explicitly describe the centralizing monoid M(mb ) with mb as its witness and then prove that it is not maximal if k > 3, contrary to the case for k = 3.