# Algebra Seminar talk

2010-03-19

Hajime **Machida***Centralizers of Monoids and Monoids of Centralizers*

Abstract:

Let A be a non-empty set. Denote by O_{A}^{(n)} the set of
n-variable functions defined on A and by O_{A} the union of O_{A}^{(n)} for all n>0. For a subset F of O_{A} the *centralizer* F^{*} of F is the set of functions in O_{A} which commute
with all functions in F.

First, we consider monoids of unary functions, that is, non-empty subsets of O

_{A}^{(1)}containing the identity and being closed under composition. We determine centralizers of all monoids which contain the symmetric group.- Let
**A**= ( A;F) be an algebra. The unary part of the centralizer F^{*}of F is exactly the set of endomorphisms of**A**}. The unary part of a centralizer is obviously a monoid with respect to composition. We call such monoid an*endoprimal monoid*. We present a lemma, called the*witness lemma*, which can be used to obtain endoprimal monoids. For a three-element set A we determine all endoprimal monoids which have subsets of O_{A}^{(1)}as their witnesses.