  This part of a multi-paper project studies the lattice properties of the
arithmetic mean ideals of B(H) introduced by Dykema, Figiel, Weiss, and
Wodzicki. We prove: the lattices of all principal ideals, of arithmetic mean or
arithmetic mean at infinity stable principal ideals or of principal ideals with
a generator that satisfies the Delta_1/2 condition, are all both upper and
lower dense in the lattice of general ideals. That is, between any ideal and an
ideal (nested above or below respectively) in one of these sublattices, lies
another ideal in that sublattice. Among the applications: a principal ideal I
is am-stable (I = I_a) if and only if any of its first order arithmetic mean
ideals are am-stable if and only if the ideal satisfies the first order
equality cancellation property: J_a = I_a implies J = I. We show that this
cancellation property can fail even for am-stable countably generated ideals.
Similar results hold for arithmetic mean at infinity ideals. Inclusion
cancellations do not hold in general even for principal ideals, but for every
ideal I there is a largest ideal I^ for which J_a contains I_a implies that J
contains I^. When I is principal, I^ too is principal. We show that I=I^ is a
strictly stronger property than am-stability. For example, for I the p < 1
power of the principal ideal J generated by diag {1/n}, I^ is the q power of J
where 1/p-1/q = 1.
