  The spectral order on $\bR$ induces a partial ordering on the manifold
$\calH_{n}$ of monic hyperbolic polynomials of degree $n$. We show that the
semigroup $\tilde{\calS}$ generated by differential operators of the form
$(1-\la \frac{d}{dx})e^{\la \frac{d}{dx}}$, $\la \in \bR$, acts on the poset
$\calH_{n}$ in an order-preserving fashion. We also show that polynomials in
$\calH_{n}$ are global minima of their respective $\tilde{\calS}$-orbits and we
conjecture that a similar result holds even for complex polynomials. Finally,
we show that only those pencils of polynomials in $\calH_{n}$ which are of
logarithmic derivative type satisfy a certain local minimum property for the
spectral order.
