Closed subsets in Coxeter groups

Question

Let $W$ be a finite or infinite Coxeter group and $\Phi^+$ the set of its positive roots.

In the paper, a subset $A$ of $\Phi^+$ is closed if for all $a, b \in A$, $r_1 a + r_2 b \in \Phi^+$ for some $r_1, r_2 \ge 0$ implies that $r_1 a + r_2 b \in A$. A subset is biclosed if it is closed and its complement is also closed.

Are the following resuls correct? Are there some references about these?

The union of two disjoint biclosed subsets of $\Phi^+$ is biclosed.

Any help would be greated appreciated！

Answer 1

The union of two disjoint biclosed sets is not biclosed in general. For a counterexample consider the root system of type $A_2$. We denote the simple roots by $\alpha$ and $\beta$.

We claim that the set $\{\alpha\}\subseteq \Phi^+$ is biclosed. It is closed since the only scalar multiple of $\alpha$ that belongs to $\Phi^+$ is $\alpha$ itself. The complement $\Phi^+\backslash\{a\}=\{\alpha+\beta,\beta\}$ is closed since \begin{align*} \left\lbrace\,r_1(\alpha+\beta)+r_2\beta\mid r_1,r_2\geq 0\,\right\rbrace\cap\Phi^{+}=\{\alpha+\beta,\beta\}. \end{align*}

Similarly, $\{\beta\}\subseteq\Phi^+$ is closed. On the other hand, the union $A=\{\alpha\}\cup\{\beta\}=\{\alpha,\beta\}$ is not biclosed. It is not closed because $\alpha+\beta\in\Phi^+\backslash A$.