= Solution
For each $g\ne1$, choose a <value in a lattice-ordered group> $P_g$, using <Zorn's lemma>. Such a value is a <prime convex lattice subgroup>. Otherwise the last construction in Question 1(i) would give two larger <convex lattice subgroups> whose intersection is $P_g$; maximality among subgroups omitting $g$ would put $g$ in both, hence in their intersection. Thus the right coset set $\Omega_g=P_g\backslash G$ is a <totally ordered set>.
Right multiplication defines an action $\rho_g(a):P_gh\mapsto P_gha$. It is well-defined, preserves the <total order>, and has inverse $\rho_g(a^{-1})$. The coset formulas from Question 1(i) show pointwise preservation of both <join> and <meet>, because
$$
P_gha\vee P_ghb=P_g(ha\vee hb)=P_gh(a\vee b),
$$
and likewise for the <meet>.
Take the disjoint union of these chains, arranged in any fixed <order sum> of the indices $g\ne1$. Let every $a\in G$ act on each component by $\rho_g(a)$. This gives a <lattice-ordered group homomorphism> into the <order automorphisms> of one chain. It is injective: a nonidentity $a$ moves the identity coset in $\Omega_a$, since $P_aa\ne P_a$. Its image is closed under pointwise <join> and <meet>, giving exactly the required <lattice-ordered permutation group>. This is the <Holland representation theorem>.
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