Homomorphisms of the lattice of slowly oscillating functions on the half-line

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