Precomplete Clones on Infinite Sets Which Are Closed under Conjugation

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. 1. Background and the result Let X be a set and denote for all n ≥ 1 the set of n-ary operations on X by O. The union O = ⋃ n≥1 O (n) is the set of all operations on X of finite arity. A clone is a subset of O which contains all projections, i.e. all functions of the form π k (x1, . . . , xn) = xk (1 ≤ k ≤ n), and which is closed under composition of functions. Ordering the clones on X by set-theoretical inclusion, one obtains a complete algebraic lattice Cl(X). We are interested in the structure of this lattice for infinite X , in which case it has cardinality 2 |X| . We call a clone precomplete or maximal iff it is a dual atom in Cl(X). The number of precomplete clones on an infinite base set equals the size of the whole clone lattice ([15]), and there is little hope to determine all of them. However, the precomplete clones which contain O have been determined on some infinite X ([2], [5]). Theorem 1. If X is countably infinite or of weakly compact cardinality, then there are exactly two precomplete clones Pol(T1) and Pol(T2) above O . For most other cardinalities of X , the number of precomplete clones above O is 2 |X| , so in those cases it seems impossible to find them all ([5]). The precomplete clones which contain the set of permutations S of the base set but not O have been determined on countably infinite X in [6], and we extended this result to all X of regular cardinality in [13]. To describe these clones, the following concept was used: For a submonoid G ⊆ O, define the clone of 1991 Mathematics Subject Classification. Primary 08A40; secondary 08A05.