Determine whether Conditions A and B hold for general Artin–Tits coverings

Determine, for proper coverings of irreducible spherical type Artin–Tits monoids, whether Condition A and Condition B hold beyond the cases established in the paper and the previously known exceptional cases.

Background

The paper studies Dehornoy structures arising from alternating normal forms in spherical type Artin–Tits monoids. Arcis and Paris introduced Conditions A and B as sufficient conditions ensuring that a covering yields a Dehornoy structure and hence a left-ordering of the associated Artin–Tits group.

The authors state that, even for Artin–Tits groups, the validity of these two conditions was known only in a few cases. The paper substantially classifies Condition A and rules out many instances of Condition B, but leaves several coverings unresolved.

References

Even for Artin-Tits groups, the question of whether or not Condition~$A$ and Condition~$B$ hold is open except for only few cases.

— Ordering, Artin--Tits group, Garside structure  (2609.04757 - Fromentin et al., 4 Sep 2026) in Section 1, Introduction

We also conjecture that Condition A is also satisfied for $i\geq 2$ (Corollary~\ref{C:ConditionAIff} cannot be applied here).

— Ordering, Artin--Tits group, Garside structure  (2609.04757 - Fromentin et al., 4 Sep 2026) in Section 4, subsection “Case of $D_r$”

It remains the $8$ following coverings with an open status and for which Condition A seem to be satisfied with respect to~$2$.

— Ordering, Artin--Tits group, Garside structure  (2609.04757 - Fromentin et al., 4 Sep 2026) in Section 4, subsection “Case of type $E_6$”

The remaing four coverings below have an open status for $$ and for $2$, regarding Condition A. It seems for all of them that Condition A is satisfied with respect to $$ and to $2$.

— Ordering, Artin--Tits group, Garside structure  (2609.04757 - Fromentin et al., 4 Sep 2026) in Section 4, subsection “Case of type $E_7$”

The following 7 coverings have an open status. For all of them Condition A seems to be satisfied with respect to $$ and so to $2$.

— Ordering, Artin--Tits group, Garside structure  (2609.04757 - Fromentin et al., 4 Sep 2026) in Section 4, subsection “Case of type $E_8$”

It remains $4$ proper and irreducible coverings in types $H_3$ with an unknown status relatively to Condition~$B(p)$.

— Ordering, Artin--Tits group, Garside structure  (2609.04757 - Fromentin et al., 4 Sep 2026) in Section 5, subsection “Type $H_3$”