The closure of a lattice-ordered permutation group

Abstract

Let \( (G,\mit\Omega) \) be an l-permutation group, with \( \mit\Omega \) a chain and \( \bar{\mit\Omega} \) its Dedekind completion. The gate completion \( (G^:,\bar{\mit\Omega}) \) consists of the elements of the automorphism group \( A(\bar{\mit\Omega}) \) which can be "gated" by elements of G, or equivalently, which respect the "tyings" (roughly, equality of stabilizer subgroups) of \( (G,\mit\Omega) \) [7].In this sequel we find that \( (G^:,\bar{\mit\Omega}) \) and its variant \( (G^{o:},\bar{\mit\Omega}) \) have order closed stabilizer subgroups, making \( G^: \) and \( G^{o:} \) completely distributive l-groups. The order closure of Gin \( A(\bar{\mit\Omega}) \) turns out to be \( G^{o:} \). Moreover, the elements of \( G^{o:} \) (and of \( G^: \)) can be readily constructed from those of G. Every \( h \in G^{o:} \) can be written as¶¶\( h = \bigvee \limits _i \bigwedge \limits _j \bigvee \limits _k g_{ijk}\qquad (g_{ijk} \in G) \)¶¶When \( \mit\Omega \) is countable, this can be improved to say that \( h = \bigvee \limits _i \bigwedge \limits _j g_{ij}\qquad (g_{ij} \in G) \)¶¶These results illuminate several of the standard completions of abstract completely distributive l-groups. The proofs use the structure theory of l-permutation groups to focus on the constraints imposed by tyings.