How widely applicable is the following idea? Let f: Z x Z -> Z be a binary FUNCTION (in the sense of sets) on the integers, with the property that - for each x:Z, f(x,-) : Z -> Z is a (agrees with a unique) POLYNOMIAL, whose coefficients are functions of x; and similarly - for each y:Z, f(-,y) : Z -> Z is also a polynomial. Then f(x,y) was itself a polynomial in two variables. This generalises to a disjoint union of sets of variables, ie to functions Z^X x Z^Y -> Z that are polynomials in one set of variables for each value of the other, and vice versa. The (possible) categorical generalisation is this: Let T be a strong monad on a topos, CCC or even a symmetric monoidal closed category, and K=T0 its free algebra. Then there is a natural transformation r_X: T X ---> K^(K^X), which we suppose to be mono. (How widely is this the case?) Then the above result states that (in that case) the square (K^Y) T (X + Y) >---------> TX V | V | | | | ----- | K^Y | | r_X | K^X | V r_Y V (K^X) >--------> (K^X x K^Y) TY K is a pullback (in fact, an intersection). I am primarily concerned with the case where T encodes the theory of frames over either Set or Dcpo, though if the result extends to commutative rings or other algebraic theories, that would be very interesting too. Paul Taylor