All Centralising Monoids with Majority Witnesses On a Four-element Set

Extending previous work by Goldstern, Machida and Rosenberg, we study unary parts of centraliser clones on a four-element set, so-called centralising monoids. We describe and count all centralising monoids on the set {0, 1, 2, 3} having majority operations as witnesses, and we list the inclusion maximal proper submonoids of the full transformation monoid among them. As it turns out only 147 out of 1715 monoids have this property.