  We derive by lattice theory a universal quantum certainty relation for
arbitrary $M$ observables in $N$-dimensional system, which provides a
state-independent maximum lower bound on the direct-sum of the probability
distribution vectors (PDVs) in terms of majorization relation. While the utmost
lower bound coincides with $(1/N,...,1/N)$ for any two orthogonal bases, the
majorization certainty relation for $M\geqslant3$ is shown to be nontrivial.
The universal majorization bounds for three mutually complementary observables
and a more general set of observables in dimension-2 are achieved. It is found
that one cannot prepare a quantum state with PDVs of incompatible observables
spreading out arbitrarily. Moreover, we obtain a complementary relation for the
quantum coherence as well, which characterizes a trade-off relation of quantum
coherence with different bases.
