Precomplete clones on infinite sets which are closed under conjugation

Abstract

We show that on an infinite set, there exist no other precomplete clones closed under conjugation except those which contain all permutations. Since on base sets of some infinite cardinalities, in particular on countably infinite ones, the precomplete clones containing the permutations have been determined, this yields a complete list of the precomplete conjugation-closed clones in those cases. In addition, we show that there exist no precomplete submonoids of the full transformation monoid which are closed under conjugation except those which contain the permutations; the monoids of the latter kind are known.