  We consider semiflows in general Banach spaces motivated by monotone cyclic
feedback systems or differential equations with integer-valued Lyapunov
functionals. These semiflows enjoy strong monotonicity properties with respect
to cones of high ranks, which imply order-related structures on the
$\omega$-limit sets of precompact semi-orbits. We show that for a
pseudo-ordered precompact semi-orbit the $\omega$-limit set $\Omega$ is either
ordered, or is contained in the set of equilibria, or possesses a certain
ordered homoclinic property. In particular, we show that if $\Omega$ contains
no equilibrium, then $\Omega$ itself is ordered and hence the dynamics of the
semiflow on $\Omega$ is topologically conjugate to a compact flow on
$\mathbb{R}^k$ with $k$ being the rank. We also establish a
Poincar\'{e}-Bendixson type Theorem in the case where $k=2$. All our results
are established without the smoothness condition on the semiflow, allowing
applications to such cellular or physiological feedback systems with piecewise
linear vector fields and to such infinite dimensional systems where the
$C^1$-Closing Lemma or smooth manifold theory has not been developed.
