  We conjecture a strong property for the up and down maps U and D in an
r-differential poset: DU+tI and UD+tI have Smith normal forms over Z[t]. In
particular, this would determine the integral structure of the maps U, D, UD,
DU, including their ranks in any characteristic.
  As evidence, we prove the conjecture for the Young-Fibonacci lattice YF
studied by Okada and its r-differential generalizations Z(r), as well as
verifying many of its consequences for Young's lattice Y and the r-differential
Cartesian products Y^r.
