Monoidal intervals of clones on infinite sets

AbstractLet X be an infinite set of cardinality κ. We show that if L is an algebraic and dually algebraic distributive lattice with at most 2κ completely join irreducibles, then there exists a monoidal interval in the clone lattice on X which is isomorphic to the lattice 1+L obtained by adding a new smallest element to L. In particular, we find that if L is any chain which is an algebraic lattice, and if L does not have more than 2κ completely join irreducibles, then 1+L appears as a monoidal interval; also, if λ⩽2κ, then the power set of λ with an additional smallest element is a monoidal interval. Concerning cardinalities of monoidal intervals these results imply that there are monoidal intervals of all cardinalities not greater than 2κ, as well as monoidal intervals of cardinality 2λ, for all λ⩽2κ