Algebraic properties of totally irreducible elements of clone lattices

Totally join- (meet-) irreducible elements are important for algebraic lattices since the join-(meet-) generates them. We study such elements for clone lattices, particularly for the clone lattice on a k-element universe. We show that in this case the set of totally join- (meet-) irreducible clones is countable; in particular that there are 2/sup k-1/(2/sup k/-2) countable descending chains of such clones.