The incident facet and chamber poset

For the facets, incidence and localized chambers of Facet incidence and localized chamber signs, define $P(\mathcal{A})=\{(F,C):F\in \mathcal{F},C\in \mathcal{C},F⪯C\}$. Give it the relation $(F,C)\le (G,D)$ if $F⪯G$ and $C_F\subseteq D_G$. A partial order has the reflexivity, antisymmetry and transitivity axioms of Partial order and partially ordered set. This is Paris's ${\mathrm{Sal}}_0(\mathcal{A})$, with the same direction of ordering.

Sign facets of a cone arrangement

Facet incidence and localized chamber signs

Partial order and partially ordered set

Given: the incident pairs and relation in the Definition; the facets, chambers and signs of Sign facets of a cone arrangement; the incidence and localized chamber signs of Facet incidence and localized chamber signs; and the order axioms of Partial order and partially ordered set.

1.1 For $F⪯G$, the containment $C_F\subseteq D_G$ holds exactly when the signs of $C,D$ agree on ${\mathcal{A}}_G$: necessity follows by evaluating any point of the nonempty $C_F$, and sufficiency follows because ${\mathcal{A}}_G\subseteq {\mathcal{A}}_F$. [given]

1.2 Reflexivity follows from reflexivity of incidence and containment. For transitivity, if $(F,C)\le (G,D)\le (K,L)$, then the signs of $C,D$ agree on ${\mathcal{A}}_G$, hence on ${\mathcal{A}}_K$, and those of $D,L$ agree on ${\mathcal{A}}_K$. Incidence is transitive, so $(F,C)\le (K,L)$. [given, step 1.1]

1.3 For antisymmetry, opposite inequalities force $F=G$ and $C_F=D_F$; the last assertion of the incidence lemma gives $C=D$. Thus the displayed relation satisfies all three partial-order axioms. [given, step 1.2] ∎

Construction compared with Paris §3.1, printed pp.385–386, including the displayed order on ${\mathrm{Sal}}_0(\mathcal{A})$. Its elementary order verification is supplied here.