The closure of a lattice-ordered permutation group (Q1272115)

Let \(\Omega \) be a chain, \(\overline {\Omega }\) its Dedekind completion, and \((G,\Omega)\) a lattice-ordered permutation group (an \(l\)-permutation group). The authors study some completions of \((G,\Omega)\) in the automorphism group \(A(\overline {\Omega })\). For example, they investigate the gate completion consisting of the elements of \(A(\overline {\Omega })\) which can be gated by elements of \(G\), or equivalently, which respect the tyings of \((G,\Omega)\). The results of the paper illuminate several of the standard completions of abstract completely distributive \(l\)-groups using the structure theory of \(l\)-permutation groups.