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Maximal Centralizing Monoids Related to Minimal Clones
Hajime Machida
Let 𝐴 be a finite set. A centralizing monoid 𝑀 on 𝐴 is a set of unary operations which commute with some set 𝐹 of operations on 𝐴. 𝐹 is called a witness of 𝑀. A maximal centralizing monoid has a witness which is a singleton set consisting of a minimal operation.
On a three-element case, i.e., |𝐴| = 3, a centralizing monoid is maximal if and only if it has a constant operation or a majority minimal operation as its witness.
In this article, we aim at extending the above property to any finite set 𝐴 with |𝐴| ≥ 3. A new proof is presented of the fact that a centralizing monoid having a constant operation as its witness is maximal. For the majority operation witnesses, we prove that, for two kinds of majority minimal operations, a centralizing monoid having one of those operations as its witness is maximal, except one case for |𝐴| = 4.
Keywords: Clone, centralizer, centralizing monoid, minimal clone