A Study on Centralizing Monoids with Majority Operation Witnesses.

A centralizing monoidM 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 m_b, on a k-element set for any k = 3. We explicitly describe the centralizing monoid M(m_b) with m_b as its witness and then prove that it is not maximal if k > 3, contrary to the case for k = 3. [ABSTRACT FROM AUTHOR]