From the authors’ abstract: In an \(\ell \)-permutation group \((G,\Omega)\), with \(\Omega \) a chain and \(\overline \Omega \) its Dedekind completion, the coincidence of two stabilizer subgroups \(G_\delta =G_\lambda\) \((\delta , \lambda \in \overline \Omega)\) yields a map \(\delta g\to \lambda g\) \((g\in G)\) from \(\delta G\) to \(\lambda G\), and this map commutes with all the elements of \(G\). Roughly speaking, a tying is such a map. We show that the permutations of \(\overline \Omega \) which commute with the tyings are exactly those in the closure of \(G\) in the full automorphism group \(A(\overline \Omega)\) with respect to the coarse stabilizer topology. We term this closure the gate completion of \(G\), written \(G^:\). We show that each \(o\)-primitive component of \(G^:\) consists of those permutations of the closure of the corresponding \(G\) component which respect the orbits of the points which are “tied” to nonsingleton \(o\)-blocks. Finally, we show that any two representations of the same lattice-ordered group which are complete and without dead segments give rise to the same \(G^:\), and that in this case \(G^:\) is the \(\alpha\)-completion of \(G\).