  If $x$ and $y$ are roots in the root system with respect to the standard
(Tits) geometric realization of a Coxeter group $W$, we say that $x$
\emph{dominates} $y$ if for all $w\in W$, $wy$ is a negative root whenever $wx$
is a negative root. We call a positive root \emph{elementary} if it does not
dominate any positive root other than itself. The set of all elementary roots
is denoted by $\E$. It has been proved by B. Brink and R. B. Howlett (Math.
Ann. \textbf{296} (1993), 179--190) that $\E$ is finite if (and only if) $W$ is
a finite-rank Coxeter group. Amongst other things, this finiteness property
enabled Brink and Howlett to establish the automaticity of all finite-rank
Coxeter groups. Later Brink has also given a complete description of the set
$\E$ for arbitrary finite-rank Coxeter groups (J. Algebra \textbf{206} (1998)).
However the set of non-elementary positive roots has received little attention
in the literature. In this paper we answer a collection of questions concerning
the dominance behaviour between such non-elementary positive roots. In
particular, we show that for any finite-rank Coxeter group and for any
non-negative integer $n$, the set of roots each dominating precisely $n$ other
positive roots is finite. We give upper and lower bounds for the sizes of all
such sets as well as an inductive algorithm for their computation.
