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 OA(n) the set of n-variable functions defined on A and by OA the union of OA(n) for all n>0. For a subset F of OA the centralizer F* of F is the set of functions in OA which commute with all functions in F.

  1. First, we consider monoids of unary functions, that is, non-empty subsets of OA(1) containing the identity and being closed under composition. We determine centralizers of all monoids which contain the symmetric group.

  2. 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 OA(1) as their witnesses.