The Gate Completion G: of a Lattice-Ordered Permutation Group

McCleary

1997

Algebra univers.

In an l-permutation group (G, d), with d a chain and d ( its Dedekind completion, the coincidence of two stabilizer subgroups G l =G u (l, u d ( ) yields a map lg ug (g G) from lG to uG, and this map commutes with all the elements of G. Roughly speaking, a tying is such a map. We show that the permutations of d ( which commute with the tyings are exactly those in the closure of G in the full automorphism group A(d ( ) 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 h-completion of G.

Section: The Proof Of Theorem 31supporting

confidence: 50%

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…In the preview [7] (G, d) was often assumed for simplicity to have convex orbits. G was defined just as it is here, except that it acted just on dG rather than on all of d ( .…”

Section: This Formulation Makes It Clear Thatmentioning

confidence: 99%

Section: The Proof Of Theorem 31mentioning

confidence: 99%

Section: Orbit-gateabilitymentioning

confidence: 99%

See 3 more Smart Citations

Tyings in lattice-ordered permutation groups

McCleary

1997

Algebra univers.

Cauchy completions of MV-algebras

Georgescu

Leustean

2002

Algebra Universalis

Self Cite

An MV-convergence is a convergence on an MV-algebra which renders the operations continuous. We show that such convergences on a given MV-algebra A are exactly the restrictions of the bounded -convergences on the abelian -group in which A appears as the unit interval. Thus the theory of -convergence and Cauchy structures transfers to MV-algebras.We outline the general theory, and then apply it to three particular MV-convergences and their corresponding Cauchy completions. The Cauchy completion arising from order convergence coincides with the Dedekind-MacNeille completion of an MV-algebra. The Cauchy completion arising from polar convergence allows a tidy proof of the existence and uniqueness of the lateral completion of an MV-algebra. And the Cauchy completion arising from α-convergence gives rise to the cut completion of an MV-algebra.

The closure of a lattice-ordered permutation group

McCleary

1997

Algebra Universalis

Let (G, d) be an l-permutation group, with d a chain and d ( its Dedekind completion. The gate completion (G : , d ( ) consists of the elements of the automorphism group A(d ( ) which can be ''gated'' by elements of G, or equivalently, which respect the ''tyings'' (roughly, equality of stabilizer subgroups) of (G, d) [7]. In this sequel we find that (G : , d ( ) and its variant (G o: , d ( ) have order closed stabilizer subgroups, making G : and G o: completely distributive l-groups. The order closure of G in A(d ( ) 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 G o: can be written asWhen d is countable, this can be improved to say thatThese 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.