All meets and joins of the right weak order of $A_2$ ($S_3$), with the left order and the inversion sets compared

The six elements: the type-$A$ isomorphism and inversion-length formula [F3,F4] give $\mathrm{\mid }W\mathrm{\mid }=6$, $\mathrm{\ell }(s)=\mathrm{\ell }(t)=1$, $\mathrm{\ell }(st)=\mathrm{\ell }(ts)=2$ and $\mathrm{\ell }(sts)=3$. The values $1,s,t,st,ts,sts$ are distinct: $1$ has length $0$ by [F2], the isomorphism separates $s$ and $t$, elements of different lengths are distinct, and $st=ts$ would imply $sts=t$ after right multiplication by $s$, contradicting their lengths. From the presentation [F1], $(st{)}^3=1$ and $s^2=t^2=1$; hence $(sts)(tst)=1$, while $tst$ is an involution. Therefore $sts=tst=:w_0$, and the six displayed elements exhaust $W$.

The $A_2$ root set and reflection actions: put $c:=\cos (\pi \mathrm{/}3)$. By [F18,F19], $2c^2-1=\cos (2\pi \mathrm{/}3)=-c$, so $(2c-1)(c+1)=0$. Since $\pi \mathrm{/}3<\pi \mathrm{/}2$, [F20] and [F21] give $c=\cos (\pi \mathrm{/}3)>\cos (\pi \mathrm{/}2)=0$, hence $c=1\mathrm{/}2$. From the Coxeter form and reflection formula [F11,F12], $\rho (s)e_s=-e_s$, $\rho (s)e_t=e_t+e_s$, $\rho (t)e_t=-e_t$, and $\rho (t)e_s=e_s+e_t$; thus $\rho (s)(e_s+e_t)=e_t$ and $\rho (t)(e_s+e_t)=e_s$. These actions and linearity show that $\mathcal{R}:=\{\pm e_s,\pm e_t,\pm (e_s+e_t)\}$ is invariant under both generators. Since $W$ is generated by $S$ [F2] and $\rho$ is a homomorphism [F13], every $\rho (w)$ preserves $\mathcal{R}$. The root-orbit definition [F14] gives $\mathrm{\Phi }\subseteq \mathcal{R}$; conversely $e_s,e_t$ are roots, $-e_s=\rho (s)e_s$, $-e_t=\rho (t)e_t$, $e_s+e_t=\rho (s)e_t$, and $-(e_s+e_t)=\rho (s)\rho (t)e_t$, so $\mathrm{\Phi }=\mathcal{R}$. The positive cone and sign theorem [F15] then give ${\mathrm{\Phi }}_{+}=\{{\alpha }_s,{\alpha }_t,{\alpha }_s+{\alpha }_t\}$, where ${\alpha }_s=e_s$ and ${\alpha }_t=e_t$.

Covers: the cover characterization [F7] says a right cover is right multiplication by a simple generator with length rise one. Multiplying the six values of step 1.1 by $s,t$ gives the rises $1s=s$, $1t=t$, $st=s\cdot t$, $ts=t\cdot s$, $w_0=st\cdot s$, and $w_0=ts\cdot t$; each raises length by one by [F4]. The other six products are $ss=1$, $tt=1$, $(st)t=s$, $(ts)s=t$, $w_{0}s=st$, and $w_{0}t=ts$, none of which raises length. Hence the right covers are exactly the six displayed in the statement.

The asymmetry: $st=s\cdot t$ and $\mathrm{\ell }(st)=\mathrm{\ell }(s)+\mathrm{\ell }(t)$, so $s{\le }_{R}st$ by [F5]. If $s{\le }_{L}st$, [F6] gives $st=xs$ and $\mathrm{\ell }(st)=\mathrm{\ell }(x)+\mathrm{\ell }(s)$; right multiplication by $s$ gives $x=sts=w_0$, contradicting $2=\mathrm{\ell }(st)=\mathrm{\ell }(x)+1=4$. Likewise $st=s\cdot t$ with $\mathrm{\ell }(st)=\mathrm{\ell }(s)+\mathrm{\ell }(t)$ gives $t{\le }_{L}st$, while $t{\le }_{R}st$ would force $x=t^{-1}st=tst=w_0$ and $2=\mathrm{\ell }(st)=1+3$, impossible. Inversion is an order isomorphism by [F9], consistent with these comparisons.

The six inversion sets: by the definition [F16] and prefix-root formula [F17], the reduced expressions from step 1.1 and the roots from step 1.2 give $N(1)=\mathrm{\varnothing }$, $N(s^{-1})=N(s)=\{{\alpha }_s\}$, $N(t^{-1})=N(t)=\{{\alpha }_t\}$, $N((st{)}^{-1})=N(ts)=\{{\alpha }_s,\rho (s){\alpha }_t\}=\{{\alpha }_s,{\alpha }_s+{\alpha }_t\}$, and $N((ts{)}^{-1})=N(st)=\{{\alpha }_t,\rho (t){\alpha }_s\}=\{{\alpha }_t,{\alpha }_s+{\alpha }_t\}$. For $w_0=sts=tst$, the same formula gives $N(w_0^{-1})=\{{\alpha }_s,\rho (s){\alpha }_t,\rho (st){\alpha }_s\}=\{{\alpha }_s,{\alpha }_s+{\alpha }_t,{\alpha }_t\}={\mathrm{\Phi }}_{+}$, since $\rho (st){\alpha }_s=\rho (s)({\alpha }_s+{\alpha }_t)={\alpha }_t$.

The right order: the cover chains of step 2.1 and the chain characterization in [F7] give exactly the $17$ relations: $1{\le }_{R}v$ for all six $v$; $s{\le }_{R}s,st,w_0$; $t{\le }_{R}t,ts,w_0$; $st{\le }_{R}st,w_0$; $ts{\le }_{R}ts,w_0$; and $w_0{\le }_R{w}_0$. The remaining $19$ of the $36$ ordered pairs do not satisfy $u{\le }_{R}v$; eight have incomparable entries, and eleven are reversals of strict comparisons. The down-sets are $\downarrow 1=\{1\}$, $\downarrow s=\{1,s\}$, $\downarrow t=\{1,t\}$, $\downarrow st=\{1,s,st\}$, $\downarrow ts=\{1,t,ts\}$ and $\downarrow w_0=W$; the up-sets are $\uparrow 1=W$, $\uparrow s=\{s,st,w_0\}$, $\uparrow t=\{t,ts,w_0\}$, $\uparrow st=\{st,w_0\}$, $\uparrow ts=\{ts,w_0\}$ and $\uparrow w_0=\{w_0\}$. The minimum $1$ and poset properties used here are in [F9].

The order isomorphism: inversion is an order isomorphism by [F9], so left weak order is the image of the right order under $w\mapsto w^{-1}$. In particular, $s{\le }_{R}st$ but $s\mathrm{̸}{\le }_{L}st$, while $t{\le }_{L}st$ but $t\mathrm{̸}{\le }_{R}st$, as shown in step 2.2.

Meets: intersections of down-sets from step 3.1 are $\{1\}$ for the four cross pairs $(s,t),(s,ts),(st,t),(st,ts)$; they are the down-set of the smaller element for pairs in one chain, and the down-set of the other element when one is $w_0$. By [A1] their greatest elements are the greatest common lower bounds, so the cross-pair meets are $1$, the meet along a chain is its smaller element, and $w_0\wedge v=v$ for $v\in \{s,t,st,ts\}$. Each listed value lies below both elements and dominates every common lower bound, the universal property in [F22]. For the empty meet every element is a lower bound, and the maximum $w_0$ gives $\bigwedge \mathrm{\varnothing }=w_0$, in agreement with [F10].

Joins: intersections of up-sets from step 3.1 are $\{w_0\}$ for the four cross pairs $(s,t),(s,ts),(st,t),(st,ts)$ and for every pair containing $w_0$; on a chain, the intersection is the up-set of the larger element. By [A1] their least elements are the least common upper bounds, so every cross-pair join and every join involving $w_0$ is $w_0$, and joins along a chain are its larger element. Each listed value is an upper bound and lies below every common upper bound, the universal property in [F22]. For the empty join every element is an upper bound, and the minimum $1$ gives $\bigvee \mathrm{\varnothing }=1$, in agreement with [F10].

The criterion on all $36$ pairs: write $\gamma ={\alpha }_s+{\alpha }_t$. The six sets of step 2.3 are $\mathrm{\varnothing }$, $\{{\alpha }_s\}$, $\{{\alpha }_t\}$, $\{{\alpha }_s,\gamma \}$, $\{{\alpha }_t,\gamma \}$, and $\{{\alpha }_s,{\alpha }_t,\gamma \}$. Their inclusion relations consist of the six reflexive pairs; the five strict inclusions from $\mathrm{\varnothing }$ to every nonempty set; the four inclusions from each singleton to its containing intermediate set and to the full set; and the two inclusions from the intermediate sets to the full set, for $17$ relations total. The remaining $19$ ordered pairs do not satisfy the directed inclusion $N(u^{-1})\subseteq N(v^{-1})$. Comparing these $17$ inclusions and $19$ failures with the corresponding right-order relations and failures from step 3.1 proves, in both directions, $u{\le }_{R}v⟺N(u^{-1})\subseteq N(v^{-1})$ on all $36$ pairs by [F8]. No Choice is used.