Weak order is a meet-semilattice, finite Coxeter groups are lattices, and joins of simple reflections exist exactly for finite parabolics

Clause (1): the complete meet-semilattice and bounded-join assertions in ${\le }_R$ are exactly [F3] and [F4]. Inversion is an order isomorphism by [F2], so for every nonempty $A\subseteq W$ it transports meets of $A^{-1}:=\{a^{-1}:a\in A\}$ to meets of $A$ in ${\le }_L$, and transports existence and values of joins in the same way. Thus clause (1) holds in both orders.

Clause (2): assume $W$ is finite, with longest element $w_0$ from [F22]. For every $w\in W$, applying [F11] to $w^{-1}$ and using [F8] gives $\mathrm{\ell }(w^{-1}{w}_0)=\mathrm{\ell }(w_0)-\mathrm{\ell }(w^{-1})=\mathrm{\ell }(w_0)-\mathrm{\ell }(w)$; hence $w_0=w(w^{-1}{w}_0)$ is length-additive, so $w{\le }_R{w}_0$ by [F1]. Applying this to $w^{-1}$ and then using inversion and $w_0^{-1}=w_0$ from [F12] shows $w{\le }_L{w}_0$ as well. Thus $w_0$ is a maximum in both orders, and [F2] gives their minimum $1$. Every nonempty $A\subseteq W$ is bounded above by $w_0$, so [F3] and [F4] give its join and meet in each order. For $A=\mathrm{\varnothing }$, every element is both an upper and a lower bound, so the maximum and minimum give $\bigwedge \mathrm{\varnothing }=w_0$ and $\bigvee \mathrm{\varnothing }=1$ in both orders by [F18] and [F20]. The two orders are lattices by [F16].

Clause (3), $(a)\Rightarrow (b)$, and minimality of $w_0(J)$: if $J=\mathrm{\varnothing }$, then $W_J=\{1\}$ and $w_0(J)=1$, which is an upper bound of $J$ and lies below every $w\in W$. Now assume $J\mathrm{\ne }\mathrm{\varnothing }$ and $W_J$ is finite, with longest element $w_0(J)$ from [F23]. For each $s\in J$, [F13] and [F14] give $\mathrm{\ell }(s{w}_0(J))=\mathrm{\ell }((s{w}_0(J){)}^{-1})=\mathrm{\ell }(w_0(J)s)=\mathrm{\ell }(w_0(J))-\mathrm{\ell }(s)=\mathrm{\ell }(w_0(J))-1$, using $w_0(J{)}^2=1$ from [F13], $s^2=1$ from [F17], and length invariance under inversion from [F8]. Thus $w_0(J)=s(s{w}_0(J))$ is length-additive, so $s{\le }_R{w}_0(J)$ by [F1] and $w_0(J)$ is an upper bound of $J$. To prove that boundedness forces finiteness and minimality, let $w$ be any upper bound of $J$, with no finiteness assumption on $W_J$. For each $s\in J$, [F1] gives $w=sx$ with $\mathrm{\ell }(w)=\mathrm{\ell }(s)+\mathrm{\ell }(x)$; [F17] gives $x=sw$, so [F14] implies $\mathrm{\ell }(sw)=\mathrm{\ell }(w)-1$ and [F10] gives $s\in D_L(w)$. Factor $w=w_{J}d$ uniquely as in [F7], with $w_J\in W_J$, $d\in {}^{J}W$, and $\mathrm{\ell }(vd)=\mathrm{\ell }(v)+\mathrm{\ell }(d)$ for every $v\in W_J$. Then $s{w}_J\in W_J$ and $\mathrm{\ell }(s{w}_J)+\mathrm{\ell }(d)=\mathrm{\ell }(s{w}_{J}d)=\mathrm{\ell }(sw)<\mathrm{\ell }(w)=\mathrm{\ell }(w_J)+\mathrm{\ell }(d)$, so $\mathrm{\ell }(s{w}_J)<\mathrm{\ell }(w_J)$ for every $s\in J$. By [F5], $W_J$ is finite and $w_J=w_0(J)$. Hence $w=w_0(J)d$ is length-additive, so $w_0(J){\le }_{R}w$ by [F1]: it lies below every upper bound of $J$, and boundedness of $J$ forces $W_J$ finite.

Clause (3) and the join value: if $J=\mathrm{\varnothing }$, then $\bigvee J=1=w_0(J)$ because $1$ is the minimum in both orders, and conditions (a)–(c) all hold. Suppose $J\mathrm{\ne }\mathrm{\varnothing }$. If $J$ has an upper bound, step 1.3 gives that $W_J$ is finite and that $w_0(J)$ is an upper bound below every upper bound. By [F4], the join exists and is the meet of the nonempty set $U(J)$ of upper bounds; therefore $\bigvee J=w_0(J)$. Conversely, if $W_J$ is finite then step 1.3 supplies an upper bound, and if the join exists then it is itself an upper bound by [F18]. This proves the equivalence and join value in ${\le }_R$; if $W_J$ is infinite, the equivalence shows that $J$ has no upper bound and no join.

The left-order half of clauses (1) and (3): inversion is an order isomorphism by [F2]. By [F17], each $s\in S$ is an involution, so inversion fixes $J$ pointwise and transports its upper bounds, joins and boundedness in one order to those in the other. By [F13], $w_0(J{)}^{-1}=w_0(J)$, so the join value in the left order is also $w_0(J)$.

Clause (4): let $S=\{s,t\}$ with $s\mathrm{\ne }t$ and $m(s,t)=\mathrm{\infty }$. By [F17], the Coxeter presentation here is $⟨s,t\mid s^2=t^2=1⟩$, the standard infinite dihedral presentation. By [F15], $st$ has infinite order; if $(st{)}^i=(st{)}^j$ for integers $i\mathrm{\ne }j$, then $(st{)}^{i-j}=1$, a contradiction, so these powers are pairwise distinct and $W$ is infinite. Since $W_{\{s,t\}}=W$, clause (3) shows $\{s,t\}$ has no upper bound in either weak order; consequently it has no join in either order by [F18] and [F20]. To prove incomparability, if $s{\le }_{R}t$ then [F1] gives $t=sx$ and $\mathrm{\ell }(t)=\mathrm{\ell }(s)+\mathrm{\ell }(x)$. Since $\mathrm{\ell }(s)=\mathrm{\ell }(t)=1$ by [F14], [F6] yields $x=1$, contradicting $s\mathrm{\ne }t$; the same argument with $s,t$ exchanged excludes $t{\le }_{R}s$. For ${\le }_L$, [F21] gives the factorizations $t=xs$ or $s=xt$, and the same length calculation excludes both comparisons. No Axiom of Choice is used: the arbitrary-subset meet assertion is supplied by [F3], and [F19] records that its construction makes only finitely many choices. No completeness or lattice property beyond (1) is claimed for infinite $W$.