  We study the space $H(\SO)$ of all homomorphisms of the vector lattice of all
slowly oscillating functions on the half-line $\HH=[0,\infty)$. In contrast to
the case of homomorphisms of uniformly continuous functions, it is shown that a
homomorphism in $H(\SO)$ which maps the unit to zero must be zero-homomorphism.
Consequently, we show that the space $H(\SO)$ without zero-homomorphism is
homeomorphic to $\HH\times (0, \infty)$. By describing a neighborhood base of
zero-homomorphism, we show that $H(\SO)$ is homeomorphic to the space
$\HH\times (0, \infty)$ with one point added.
