Relatively uniformly continuous (ruc) semigroups were recently introduced and
studied by Kandi\'c, Kramar-Fijav\v{z}, and the second-named author, in order
to make the theory of one-parameter operator semigroups available in the
setting of vector lattices, where no norm is present in general.
In this article, we return to the more standard Banach lattice setting -
where both ruc semigroups and C0-semigroups are well-defined concepts - and
compare both notions. We show that the ruc semigroups are precisely those
positive C0-semigroups whose orbits are order bounded for small times.
We then relate this result to three different topics: (i) equality of the
spectral and the growth bound for positive C0-semigroups; (ii) a uniform
order boundedness principle which holds for all operator families between
Banach lattices; and (iii) a description of unbounded order convergence in
terms of almost everywhere convergence for nets which have an uncountable index
set containing a co-final sequence.Comment: 13 pages. This is version 2. Minor changes compared to version