Monoidal intervals of clones on infinite sets

Let X   be an infinite set of cardinality κκ. We show that if LL is an algebraic and dually algebraic distributive lattice with at most 2κ2κ completely join irreducibles, then there exists a monoidal interval in the clone lattice on X   which is isomorphic to the lattice 1+L1+L obtained by adding a new smallest element to LL. In particular, we find that if LL is any chain which is an algebraic lattice, and if LL does not have more than 2κ2κ completely join irreducibles, then 1+L1+L appears as a monoidal interval; also, if λ⩽2κλ⩽2κ, then the power set of λλ 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 2κ2κ, as well as monoidal intervals of cardinality 2λ2λ, for all λ⩽2κλ⩽2κ.