Q:

Lattice structure in the root poset

Let $W$ be a Coxeter group with simple generators $s_1$, $s_2$, ..., $s_r$. Let $\Phi^+$ be the corresponding positive root system, with $\alpha_i$ the positive root corresponding to $s_i$. Bjorner and Brenti, Combinatorics of Coxeter Groups, Chapter Four define the root poset to be a partial order on $\Phi$ as follows: 
If $\beta \in \Phi$ and $s_i \beta - \beta \in \mathbb{R}_{>0} \alpha_i$, then $\beta < s_i \beta$. The root poset is then the transitive closure of this relation. 
Bjorner and Brenti, Exercise 4.15 asks: 

Is the positive root poset $(\Phi^+, \leq)$, with a bottom element appended, a meet-semilattice?

I can't find the answer to this exercise. Can someone help?
What I would actually like to know is:

Are intervals $[\beta, \gamma]$ in the root poset lattices?

Below, some bibliographic notes:

I cheated slightly above: Bjorner and Brenti, as well as the sources below, actually only order $\Phi^+$, not $\Phi$. But I see no reason not to extend the order to the negative roots.
This poset was introduced in Henrik Eriksson's PhD thesis and, independently, by Brink and Howlett:

Brink, Brigitte; Howlett, Robert B., A finiteness property and an automatic structure for Coxeter groups, Math. Ann. 296, No. 1, 179-190 (1993). ZBL0793.20036.

This poset is not the same as defining $\beta \leq \gamma$ if $\gamma - \beta$ is in the positive span of the $\alpha_i$; a condition which is also sometimes called the root poset. 
Let $\beta$ be a positive root and $t$ the corresponding reflection. Then $s_i \beta - \beta \in \mathbb{R}_{>0} \alpha_i$ if and only if $s_i$ is an inversion of $s_i t s_i$. Thus, we can define this relation in a purely Coxeter theoretic way, without mentioning root systems.

A:

The root poset for $\tilde{A_2}$ is shown in Figure 4.5 in the same reference and copied below.  One can check that the elements labelled $112$ and $221$ have both $100$ and $010$ as maximal common lower bounds, so meets don't exist in general.  If you take an isomorphic example higher up in this same poset you can get an interval which is not a lattice.
$\tilde{A_2}$.">

