The right and left weak orders, intervals, covers, and meets and joins of subsets

Let $(S,m)$ be a finite Coxeter matrix, let $W$ be the presented group with length function $\mathrm{\ell }$ and reduced expressions (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups), and for $w\in W$ let

$D_L(w):=\{s\in S:\mathrm{\ell }(sw)<\mathrm{\ell }(w)\},D_R(w):=\{s\in S:\mathrm{\ell }(ws)<\mathrm{\ell }(w)\}$

be the left and right descent sets already fixed in Standard parabolic subgroups, descent-free one- and two-sided representatives, parabolic and reflection subgroups (2). Let $T$, $\mathrm{\Phi }={\mathrm{\Phi }}_{+}\bigsqcup {\mathrm{\Phi }}_{-}$ and the inversion sets $N(w)=\{\alpha \in {\mathrm{\Phi }}_{+}:\rho (w)\alpha \in {\mathrm{\Phi }}_{-}\}$ be the reflection set, the signed root system and the inversion sets of The canonical reflection homomorphism, roots, reflections, and the positive cone and The geometric inversion set $N(w)$ of an element of a Coxeter group.

(1) Right and left weak order. Define two relations on $W$ by

$u{\le }_{R}v⟺v=ux\text{ for some }x\in W\text{ with }\mathrm{\ell }(v)=\mathrm{\ell }(u)+\mathrm{\ell }(x),$

$u{\le }_{L}v⟺v=xu\text{ for some }x\in W\text{ with }\mathrm{\ell }(v)=\mathrm{\ell }(u)+\mathrm{\ell }(x)\mathrm{.}$

These are the right weak order and the left weak order on $W$. Inversion relates them definitionally: $u{\le }_{R}v⟺u^{-1}{\le }_L{v}^{-1}$, because $v=ux$ with $\mathrm{\ell }(v)=\mathrm{\ell }(u)+\mathrm{\ell }(x)$ is equivalent to $v^{-1}=x^{-1}{u}^{-1}$ with $\mathrm{\ell }(v^{-1})=\mathrm{\ell }(u^{-1})+\mathrm{\ell }(x^{-1})$, using $\mathrm{\ell }(y)=\mathrm{\ell }(y^{-1})$ (Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification (3)). Write $u{<}_{R}v$ for $u{\le }_{R}v$ with $u\mathrm{\ne }v$.

(2) Intervals, covers and bounded subsets. For $u,v\in W$ define $[u,v{]}_R:=\{w\in W:u{\le }_{R}w\text{ and }w{\le }_{R}v\}$ and $[u,v{]}_L:=\{w\in W:u{\le }_{L}w\text{ and }w{\le }_{L}v\}$; these are the right and left intervals. An element $v$ covers $u$ in ${\le }_R$, written $u{⋖}_{R}v$, when $u{<}_{R}v$ and there is no $w\in W$ with $u{<}_{R}w{<}_{R}v$; define $u{⋖}_{L}v$ in the same way using ${\le }_L$. A subset $A\subseteq W$ is bounded above in ${\le }_R$ if there exists $y\in W$ with $a{\le }_{R}y$ for every $a\in A$, and bounded below if there exists $y\in W$ with $y{\le }_{R}a$ for every $a\in A$. Define upper and lower bounds in ${\le }_L$ by replacing ${\le }_R$ with ${\le }_L$.

(3) Meets and joins of subsets. Let $A\subseteq W$ and $z\in W$. The element $z$ is a right upper bound of $A$ if $a{\le }_{R}z$ for every $a\in A$; it is a right join (least upper bound) if it is a right upper bound and $z{\le }_{R}y$ for every right upper bound $y$ of $A$. Dually, $z$ is a right lower bound if $z{\le }_{R}a$ for every $a\in A$, and a right meet (greatest lower bound) if it is a right lower bound and $u{\le }_{R}z$ for every right lower bound $u$ of $A$. Define left upper and lower bounds, meets, and joins by replacing ${\le }_R$ with ${\le }_L$. Whenever the relevant meet or join is unique, write it as $\bigwedge A$ or $\bigvee A$ in the order under discussion; for $A=\{u,v\}$ write $u\wedge v$ or $u\vee v$. A meet or join, when it exists, is unique in either order: any two meets (respectively joins) bound one another, so antisymmetry gives equality. The partial-order property needed here is the recorded well-definedness justifier. This definition asserts no existence of meets or joins for any specified subset.

(4) Abstentions. This definition records ${\le }_R,{\le }_L$ and the interval and bound vocabulary; it asserts no further property. In particular, it does not assert that either relation is a partial order (Partial order and partially ordered set), that covers have the form $v=us$, that any meet or join exists, or that $w\mapsto N(w^{-1})$ computes ${\le }_R$ by inclusion of inversion sets. No Choice is used.