The upper endpoint of a c-Cambrian fiber, interval fibers and the explicit formula u_c(w) = pi_{c^{-1}}(ww0)w0

Fix $J=S\setminus \{s\}$ and let $w=w_J\cdot {}^{J}w$ be the length-additive parabolic factorization. The prefix inversion formula in [F4] and opposition in [F5] give $N(((w{w}_0{)}_J{)}^{-1})=N((w{w}_0{)}^{-1})\cap {\mathrm{\Phi }}_{J,+}=N(w_0{w}^{-1})\cap {\mathrm{\Phi }}_{J,+}={\mathrm{\Phi }}_{J,+}\setminus (N(w^{-1})\cap {\mathrm{\Phi }}_{J,+})={\mathrm{\Phi }}_{J,+}\setminus N(w_J^{-1})$. Applying opposition inside $W_J$ gives $N(w_0(J)w_J^{-1})={\mathrm{\Phi }}_{J,+}\setminus N(w_J^{-1})=N((w_J{w}_0(J){)}^{-1})$. Both $(w{w}_0{)}_J$ and $w_J{w}_0(J)$ lie in $W_J$, so equality of their inverse inversion sets gives equality of the elements by the order criterion and antisymmetry in [F6]. Hence $(w{w}_0{)}_J=w_J{w}_0(J)$.

Suppose $s$ is final in $c$ and $\mathrm{\ell }(sw)<\mathrm{\ell }(w)$; set $q:={\pi }_c(w)$ and $v:={\pi }_{cs}(w_J)$. By [F3] and [F10], both $s$ and $v$ are $c$-sortable and lie below $w$, since $s{\le }_{R}w$ and $v{\le }_R{w}_J{\le }_{R}w$. Their join $x:=s\vee v$ is $c$-sortable by [F14] and below $w$, so $x{\le }_{R}q$ by [F11]. Thus $q{\ge }_{R}s$; the terminal cover decomposition [F12] gives $q=s\vee q_J$. The prefix $q_J$ is $cs$-sortable by [F10] and $q_J{\le }_R{w}_J$ by [F4], hence $q_J{\le }_{R}v$ by [F11] applied inside $W_J$. Therefore $q=s\vee q_J{\le }_{R}s\vee v=x$, and with $x{\le }_{R}q$ this proves ${\pi }_c(w)=s\vee {\pi }_{cs}(w_J)$.

We prove by lexicographic induction on $(\mathrm{\mid }S\mathrm{\mid },\mathrm{\ell }(x))$ that whenever $x{\le }_{R}y$ and ${\pi }_c(x)={\pi }_c(y)$, one has $u_c(x)=u_c(y)$. If $S=\mathrm{\varnothing }$, then $W=\{1\}$ and this holds; at every positive-rank pair assume it holds for all smaller measures, and fix an initial letter $s$ of $c$.

Define $\tau (w):=w{w}_0$, so $u_c=\tau \circ {\pi }_{c^{-1}}\circ \tau$. The map $\tau$ reverses right weak order: if $y=xv$ with additive length, then $x{w}_0=y{w}_0(w_0{v}^{-1}{w}_0)$ and $\mathrm{\ell }(w_0{v}^{-1}{w}_0)=\mathrm{\ell }(v)$ by [F5], so $y{w}_0{\le }_{R}x{w}_0$ by [F6]; since ${\tau }^2$ is the identity, this is an order anti-isomorphism. Therefore $u_c$ is order-preserving by [F13]. Since ${\pi }_{c^{-1}}(w{w}_0){\le }_{R}w{w}_0$ by [F9], applying $\tau$ gives $w{\le }_R{u}_c(w)$. Finally, $u_c(u_c(w))=\tau ({\pi }_{c^{-1}}({\pi }_{c^{-1}}(w{w}_0)))=\tau ({\pi }_{c^{-1}}(w{w}_0))=u_c(w)$ by idempotence in [F9].

If $s$ is final in $c$ and $\mathrm{\ell }(sw)>\mathrm{\ell }(w)$, then $s$ is initial in $c^{-1}$ and $\mathrm{\ell }(sw{w}_0)<\mathrm{\ell }(w{w}_0)$ by [F5]. The initial-letter recursion [F2] gives ${\pi }_{c^{-1}}(w{w}_0)=s{\pi }_{(scs{)}^{-1}}((sw)w_0)$, so $u_c(w)=s{u}_{scs}(sw)$, proving (1)(ii). If $s$ is initial in $c$ and $\mathrm{\ell }(sw)>\mathrm{\ell }(w)$, then $s$ is final in $c^{-1}$ and $\mathrm{\ell }(sw{w}_0)<\mathrm{\ell }(w{w}_0)$; apply step 1.2 to $c^{-1}$ and $w{w}_0$ to get ${\pi }_{c^{-1}}(w{w}_0)=s\vee {\pi }_{(sc{)}^{-1}}((w{w}_0{)}_J)$. Multiplying on the right by $w_0$ converts the join to a meet by the order reversal in step 1.4; using $(w{w}_0{)}_J=w_J{w}_0(J)$ from step 1.1 and $w_0(J{)}^2=1$ gives $u_c(w)=s{w}_0\wedge ({\pi }_{(sc{)}^{-1}}(w_J{w}_0(J))w_0)=s{w}_0\wedge (u_{sc}(w_J){}^J{w}_0)$, where $u_{sc}$ is formed inside $W_J$ with longest element $w_0(J)$ and ${}^J{w}_0=w_0(J)w_0$. This proves (1)(iii).

Continue the induction of step 1.3. Suppose $x{\le }_{R}y$ and ${\pi }_c(x)={\pi }_c(y)$. If $\mathrm{\ell }(sx)<\mathrm{\ell }(x)$, then $\mathrm{\ell }(sy)<\mathrm{\ell }(y)$ because $s{\le }_{R}x{\le }_{R}y$ by [F6]. Write $y=xv$ with additive length. Then $sy=(sx)v$ and $\mathrm{\ell }(sy)=\mathrm{\ell }(sx)+\mathrm{\ell }(v)$, so $sx{\le }_{R}sy$. The recursion [F2] gives ${\pi }_{scs}(sx)={\pi }_{scs}(sy)$; since $\mathrm{\ell }(sx)=\mathrm{\ell }(x)-1$, the induction hypothesis yields $u_{scs}(sx)=u_{scs}(sy)$. In $scs$, the letter $s$ is final and $sx,sy$ have left ascent $s$, so step 2.1(ii) gives $u_{scs}(sx)=s{u}_c(x)$ and $u_{scs}(sy)=s{u}_c(y)$; cancellation proves $u_c(x)=u_c(y)$. If instead $\mathrm{\ell }(sx)>\mathrm{\ell }(x)$ but $\mathrm{\ell }(sy)<\mathrm{\ell }(y)$, then ${\pi }_c(x)={\pi }_{sc}(x_J){\le }_{R}x$ lies outside the filter above $s$, while ${\pi }_c(y)=s{\pi }_{scs}(sy)$ lies in that filter: indeed $sy\mathrm{̸}{\ge }_{R}s$, and ${\pi }_{scs}(sy){\le }_{R}sy$ by [F9], so left multiplication by $s$ lengthens this projection by one. This contradicts ${\pi }_c(x)={\pi }_c(y)$. Thus both are left ascents. The parabolic prefix is order-preserving by [F4], so $x_J{\le }_R{y}_J$; the recursion gives ${\pi }_{sc}(x_J)={\pi }_{sc}(y_J)$, and the induction hypothesis in lower rank gives $u_{sc}(x_J)=u_{sc}(y_J)$. Formula 2.1(iii), with the same $s{w}_0$ and ${}^J{w}_0$ for both inputs, now yields $u_c(x)=u_c(y)$. This completes the comparable-pair induction.

For arbitrary $x,y$ with ${\pi }_c(x)={\pi }_c(y)=v$, [F9] gives $v{\le }_{R}x,y$ and ${\pi }_c(v)=v$. Applying the comparable-pair result of step 3.1 to $(v,x)$ and $(v,y)$ gives $u_c(x)=u_c(v)=u_c(y)$. Conversely, if $u_c(x)=u_c(y)$, then ${\pi }_{c^{-1}}(x{w}_0)={\pi }_{c^{-1}}(y{w}_0)$ by the definition of $u_c$; the forward implication just proved for arbitrary pairs, applied to $c^{-1}$, gives $u_{c^{-1}}(x{w}_0)=u_{c^{-1}}(y{w}_0)$. By definition these are ${\pi }_c(x)w_0$ and ${\pi }_c(y)w_0$, so ${\pi }_c(x)={\pi }_c(y)$. Thus the two fiber partitions agree. The forward implication and idempotence of ${\pi }_c$ also give $u_c({\pi }_c(w))=u_c(w)$. Applying this identity to $c^{-1}$ and using $u_c(w)w_0={\pi }_{c^{-1}}(w{w}_0)$ yields ${\pi }_c(u_c(w))w_0=u_{c^{-1}}(u_c(w)w_0)=u_{c^{-1}}({\pi }_{c^{-1}}(w{w}_0))=u_{c^{-1}}(w{w}_0)={\pi }_c(w)w_0$, so ${\pi }_c(u_c(w))={\pi }_c(w)$.

If $y\in [x{]}_c$, then ${\pi }_c(y)={\pi }_c(x)$, so step 4.1 gives $u_c(y)=u_c(x)$; by [F9] and step 1.4, ${\pi }_c(x)={\pi }_c(y){\le }_{R}y{\le }_R{u}_c(y)=u_c(x)$. Conversely, if ${\pi }_c(x){\le }_{R}y{\le }_R{u}_c(x)$, monotonicity [F13] and step 4.1 give ${\pi }_c(x)={\pi }_c({\pi }_c(x)){\le }_R{\pi }_c(y){\le }_R{\pi }_c(u_c(x))={\pi }_c(x)$, so ${\pi }_c(y)={\pi }_c(x)$. Thus $[x{]}_c=[{\pi }_c(x),u_c(x)]$ with the asserted endpoints; the endpoint maps are order-preserving by [F13] and step 1.4.

The classes are intervals by step 5.1, and their endpoint maps are order-preserving there; applying [F16] shows that ${\sim }_c$ is a lattice congruence. It is the same kernel relation and quotient as in [F1] and the congruence conclusion of [F18], not an identification with a different least-contraction congruence. The finite quotient consequences of [F15] give the representative-independent class operations and the lattice-homomorphic quotient map. All inductions are finite and no Choice is used.