Patrik Eklund recently asked about a "double covariant power set monad". Such a monad exists. I called it the families monad in Monads of sets, Handbook of Algebra, vol. 3...it is introduced in Example 2.12 and verified on pages 78-79. In a paper under preparation on distributive laws, joint with Philip Mulry, we show that the families monad arises by a distributive law of the power set monad P over itself. To describe that law, first establish some notation as follows. For AA in PPX a family of subsets of X, let C(AA) be the cartesian product of AA, that is, the set of all choice families x = (x_A : A in AA) with x_A in A. Let I(x) be the image of x, i.e. {x_A : A in AA}. The distributive law in question, PPA -> PPA, maps AA to {I(x) : x \in C(AA)}. Based on the general theory in Beck's original paper on distributive laws, P is a submonad of PP in two ways, A |--> {A} and A |--> { {x} : x in A}. But P is also a quotient monad. As Beck has taught us, the PP-algebras are (X,a,b) with (X,a), (X,b) P-algebras married by a commutative diagram, call it (D), involving the distributive law. Think of a, b as the supremum operation of a complete semilattice structure. Then (X,a,a) satisfies (D) so that P-algebras are a variety of PP-algebras as claimed. A more interesting example is the variety of PP-algebras of form (X,a,a^op). Here (D) is known as the "complete distributive law" in the lattice literature. Thus free such algebras exist and the free algebra generated by a cardinal k has cardinality at most 2^(2^k). 16-Feb-2005 10:53:49 -0400,3817;000000000000-00000000