Ideal Clones: Solution to a Problem of Czédli and Heindorf

Given an infinite set X and an ideal I of subsets of X, the set of all finitary operations on X which map all (powers of) I-small sets to I-small sets is a clone. In [CH01], G. Czédli and L. Heindorf asked whether or not for two particular ideals I and J on a countably infinite set X, the corresponding ideal clones were a covering in the lattice of clones. We give an affirmative answer to this question. 1. Clones and ideals Let X be an infinite set and denote the set of all n-ary operations on X by O(n). Then O := ⋃ n≥1 O (n) is the set of all finitary operations on X. A subset C of O is called a clone iff it contains all projections, i.e. for all 1 ≤ k ≤ n the function π k ∈ O (n) satisfying π k (x1, . . . , xn) = xk, and is closed under composition. The set of all clones on X, ordered by set-theoretical inclusion, forms a complete algebraic lattice Cl(X). The structure of this lattice has been subject to much investigation, many results of which are summarized in the recent survey [GP]. One such result, from [Ros76], states that there exist as many dual atoms (“precomplete clones”) in Cl(X) as there are clones (that is, 22 |X| ), suggesting that it is impossible to describe all of them (as opposed to the clone lattice on finite X, where the dual atoms are finite in number and explicitly known [Ros70]). Much more recently, a new and short proof of this fact was given in [GS02]. It was observed that given an ideal I of subsets of X, one can associate with it a clone CI consisting of those operations f ∈ O which satisfy f [A] ∈ I for all A ∈ I. The authors then showed that prime ideals correspond to precomplete clones, and that moreover the clones induced by distinct prime ideals differ, implying that there exist as many precomplete clones as prime ideals on X; the latter are known to amount to 22 |X| . The study of clones that arise in this way from ideals was pursued in [CH01], for countably infinite X. The authors concentrated on the question of which ideals induce precomplete clones, and obtained a criterion for precompleteness. In the same paper, three open problems were posed, and we provide the solution to their second problem in this article. We mention that in the article [BGHP] which is still in preparation, a theory of clones which arise from ideals is being developed. In particular, that paper contains the solution to the first problem from [CH01], which 2000 Mathematics Subject Classification. Primary 08A40; secondary 08A05.