The Complete Distributivity Of Ideal Lattices

The complete distributivity of ideal lattices is investigated. It is proved that for a complete lattice L, its ideal lattice Id(L) is completely distributive if and only if every element of L is a join of finitely co-prime elements of L. Finally, we give an counterexample that the ideal lattice of a hyperalgebraic lattice is not a hypercontinuous lattice.