  We generalize the Omori-Yau almost maximum principle of the Laplace-Beltrami
operator on a complete Riemannian manifold $M$ to a second-order linear
semi-elliptic operator $L$ with bounded coefficients and no zeroth order term.
  Using this result, we prove some Liouville-type theorems for a real-valued
$C^{2}$ function $f$ on $M$ satisfying $L f \geq F(f)+ H(|\nabla f|) $ for
real-valued continuous functions $F$ and $H$ on $\Bbb R$ such that $H(0)=0$.
