In Björner and Brenti's 'Combinatorics of Coxeter Groups' (pg 137) there is the definition of a reflection ordering for a Coxeter Group: let $(W,S)$ be a Coxeter Group with some induced root system $\Phi$. A total ordering, $\prec$, on $\Phi^+$ is a reflection ordering exactly when for all $\alpha,\beta \in \Phi^+$, $\lambda,\mu \in \mathbb{R}^+$, $$\lambda\alpha+\mu\beta \in \Phi^+ \implies \alpha \prec \lambda\alpha+\mu\beta \prec \beta \quad \quad \text{or} \quad \quad \beta\prec \lambda\alpha+\mu\beta \prec \alpha.$$
Since $\Phi^+$ is in bijection with $T$, the set of reflections, we should expect some equivalent definition in without using the definition of roots. What is it exactly?
This seems like an elementary exercise so apologies if its not really appropriate to post here.