Monoidal intervals of clones on infinite sets Original Research Article

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