Monoidal intervals of clones on infinite sets

We show that for an infinite set X, if L is a completely distributive algebraic lattice with not more completely join irreducible elements than the size of the power set of X, then there is a monoidal interval in the clone lattice on X which is isomorphic to 1+L, which is L plus a new smallest element added. Concerning cardinalities of monoidal intervals this result implies that there exist monoidal intervals of all cardinalities of at most the size of the power set of X, as well as monoidal intervals of cardinality 2^k, for all cardinals k which are not greater than the power set of X.