Two distributive right weak intervals of fully commutative elements in type $A_3$

The data for $w$. The word $(s_1,s_3,s_2)$ has heap $P_w$ on positions $1,2,3$: positions $1,3$ carry $s_1,s_2$ with $m(s_1,s_2)=3$ and positions $2,3$ carry $s_3,s_2$ with $m(s_3,s_2)=3$, so $1\prec 3$ and $2\prec 3$; positions $1,2$ carry the distinct commuting letters $s_1,s_3$ with $m(s_1,s_3)=2$, so there is no generating relation between them; because positions $1,2$ are consecutive in the original word and every generating edge increases the position, no intermediate position can lie on a path between them, so no transitive path relates them. Both heap-criterion conditions hold: every chain of $P_w$ has at most two elements, so there is no convex alternating chain of length $3$, and the covering pairs $1⋖3$, $2⋖3$ have distinct labels $(s_1,s_2)$ and $(s_3,s_2)$; by [F3] the word is reduced, $w$ is fully commutative and $P_w$ is the heap of $w$, with $\mathrm{\ell }(w)=3$. The linear extensions of $P_w$ are $(1,2,3)$ and $(2,1,3)$, since both relations force position $3$ last; their labeled words are $s_1{s}_3{s}_2$ and $s_3{s}_1{s}_2$, so $C((s_1,s_3,s_2))=\mathcal{R}(w)=\{s_1{s}_3{s}_2,s_3{s}_1{s}_2\}$ by [F2] and [F3]. The order ideals are the subsets $I$ with $3\in I\Rightarrow 1,2\in I$, namely $\mathrm{\varnothing },\{1\},\{2\},\{1,2\},\{1,2,3\}$.

The data for $u$. The word $(s_1,s_3)$ has heap $P_u$ on positions $1,2$ with labels $s_1,s_3$: since $m(s_1,s_3)=2$ and the labels are distinct, no generating relation links the two positions, so $P_u$ is the two-element antichain and it has no covering pairs. Conditions (a) and (b) of [F3] hold vacuously (there is no chain of length $3$, and no covering pair at all), so the word is reduced, $u$ is fully commutative with heap $P_u$ and $\mathrm{\ell }(u)=2$. By [F2] its reduced words are the words read from the two linear extensions $(1,2)$, $(2,1)$ of $P_u$, namely $\mathcal{R}(u)=C((s_1,s_3))=\{s_1{s}_3,s_3{s}_1\}$. The order ideals of the antichain $P_u$ are all four subsets of $\{1,2\}$.

The interval $[1,u{]}_R$. By [F4] the map $x\mapsto I(x)$ is an order isomorphism $[1,u{]}_R\to J(P_u)$, so $[1,u{]}_R$ has exactly $\mathrm{\mid }J(P_u)\mathrm{\mid }=4$ elements and is a finite distributive lattice with meet and join given by intersection and union of ideals. By [F6] the products of the prefixes of the reduced words $s_1{s}_3$ and $s_3{s}_1$, namely $1,s_1,u$ and $1,s_3,u$, lie in $[1,u{]}_R$; they are pairwise distinct because their lengths are $0,1,1,2$ and $s_1\mathrm{\ne }{s}_3$ by [F5]; hence they exhaust the four-element interval and $[1,u{]}_R=\{1,s_1,s_3,u\}$. For the ideal map: $\mathrm{\ell }(s_1)=\mathrm{\ell }(s_3)=1$ and $\mathrm{\ell }(u)=2$ by [F5] and 1.2, so $\mathrm{\mid }I(s_1)\mathrm{\mid }=\mathrm{\mid }I(s_3)\mathrm{\mid }=1$ and $\mathrm{\mid }I(u)\mathrm{\mid }=2$; computing with the reduced words $(s_1)$, $(s_3)$ and $(s_1,s_3)$ gives $I(s_1)=\{1\}$, $I(s_3)=\{2\}$ and $I(u)=\{1,2\}$, while $I(1)=\mathrm{\varnothing }$ by [F4]. Hence $s_1\wedge s_3$ is the element with ideal $\{1\}\cap \{2\}=\mathrm{\varnothing }$, namely $1$, and $s_1\vee s_3$ is the element with ideal $\{1\}\cup \{2\}=\{1,2\}$, namely $u$.

The interval $[1,w{]}_R$. By [F4] the map $x\mapsto I(x)$ is a bijection $[1,w{]}_R\to J(P_w)$, so $[1,w{]}_R$ has exactly five elements by 1.1, and it is a finite distributive lattice with meets and joins given by intersection and union of ideals. By [F6], the prefixes of the two reduced words $s_1{s}_3{s}_2$ and $s_3{s}_1{s}_2$ of 1.1 show that all five displayed elements lie in $[1,w{]}_R$. Their ideals are computed as follows: $I(1)=\mathrm{\varnothing }$ and $I(w)=P_w$ by [F4]; and, using the chains $C_{s_1}=\{1\}$, $C_{s_3}=\{2\}$, $C_{s_2}=\{3\}$ of $P_w$, the reduced words $(s_1)$ and $(s_3)$ give $I(s_1)=\{1\}$ and $I(s_3)=\{2\}$, while the reduced word $(s_1,s_3)$ of 1.2 gives $I(s_1{s}_3)=\{1,2\}$; its prefixes are those of the reduced word $s_1{s}_3{s}_2$ of $w$, so that $s_1{s}_3{\le }_{R}w$ by [F6]. Their images $\mathrm{\varnothing },\{1\},\{2\},\{1,2\},\{1,2,3\}$ are the five distinct elements of $J(P_w)$, so $[1,w{]}_R=\{1,s_1,s_3,s_1{s}_3,w\}$ and each displayed element is the product of the ideal that is its image. In particular $s_1\wedge s_3$ has ideal $\{1\}\cap \{2\}=\mathrm{\varnothing }$, so $s_1\wedge s_3=1$; $s_1\vee s_3$ has ideal $\{1\}\cup \{2\}=\{1,2\}$, so $s_1\vee s_3=s_1{s}_3$; and $s_1{s}_3\vee s_1$ has ideal $\{1,2\}\cup \{1\}=\{1,2\}$, so $s_1{s}_3\vee s_1=s_1{s}_3$.

Both intervals are finite distributive lattices by 2.1 and 2.2, illustrating F4-(3). Their heaps are the two-element antichain of 1.2 and the V-shaped poset of 1.1; the first is not a chain because its two elements are incomparable, and the second is not a chain because $1$ and $2$ are incomparable in $P_w$. So the distributivity exhibited here is not the chain case.