If $A$ is an ordered Banach algebra ordered by an algebra cone $C$, then
we reference the following problem as the `domination problem': If
$0\leq a\leq b$ and $b$ has a certain property, then does $a$ inherit
this property? We extend the analysis of this problem in the setting of
radical elements and introduce it for inessential, rank one and finite
elements. We also introduce the class of $r$-inessential operators on
Banach lattices and prove that if $S$ and $T$ are operators on a Banach
lattice $E$ such that $0\leq S\leq T$ and $T$ is $r$-inessential then
$S$ is also $r$-inessential.