Ideal clones: Solution to a problem of Czedli and Heindorf

TL;DR

This paper proves that for two specific ideals on a countably infinite set, the associated ideal clones form a covering in the clone lattice, answering a question posed by Czedli and Heindorf.

Contribution

It establishes that the ideal clones for these particular ideals are a covering in the clone lattice, providing a definitive answer to an open problem.

Findings

Ideal clones form a covering in the clone lattice for the given ideals

Affirmative answer to Czedli and Heindorf's question

Advances understanding of the structure of clone lattices

Abstract

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 a 2001 article, G. Czedli 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.