Monotone semiflows in scalar non-quasi-monotone functional differential equations

By introducing a stronger than pointwise ordering, conditions are found under which scalar functional differential equations generate monotone semiflows even if they are not quasi-monotone. Typically the maximum delay must be the smaller the more quasi-monotonicity is violated. The theory of monotone semiflows is used to show that most solutions converge to equilibrium and that stability of equilibria is essentially the same as for ordinary differential equations.