  In this paper we study continuous semigroups of positive operators on general
vector lattices equipped with the relative uniform topology $\tau_{ru}$. We
introduce the notions of strong continuity with respect to $\tau_{ru}$ and
relative uniform continuity for semigroups. These notions allow us to study
semigroups on non-locally convex spaces such as $L^p(\mathbb{R})$ for $0<p<1$
and non-complete spaces such as $Lip(\mathbb{R})$, $UC(\mathbb{R})$, and
$C_c(\mathbb{R})$. We show that the (left) translation semigroup on the real
line, the heat semigroup and some Koopman semigroups are relatively uniformly
continuous on a variety of spaces.
