Algebra Seminar talk

Mike Behrisch
Centralising monoids on finite sets

Centralising monoids on a fixed set A are the Galois closed sets of the Galois connection induced by commutation between finitary operations and unary operations on A. Stating this a bit more algebraically, they are exactly all possible endomorphism monoids of any (finitary) algebra (of any similarity type) on A. Again differently put, they are exactly the unary parts of all centraliser clones on A, which for finite A are precisely those clones that are closed under primitive-positively definable functions.