<i>a</i><sup>∗</sup>-closures of completely distributive lattice-ordered groups

Glass, A. M. W.; Holland, Walter W.; McCleary, Stephen H.

doi:10.2140/pjm.1975.59.43

Cited by 7 publications

(5 citation statements)

References 13 publications

(14 reference statements)

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“…This surprising independence of representation is a consequence of the fact that G coincides with the h-completion of G in this case (Theorem 7.6). Thus this article is closely analogous to [11], inasmuch as a considerable amount of information about completions of general l-groups is obtained by l-permutation techniques.…”

Section: This Formulation Makes It Clear Thatmentioning

confidence: 99%

Tyings in lattice-ordered permutation groups

Ball

McCleary

1997

Algebra univers.

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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.

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Section: This Formulation Makes It Clear Thatmentioning

confidence: 99%

Tyings in lattice-ordered permutation groups

Ball

McCleary

1997

Algebra univers.

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“…Then G L = H; the reader may verify this himself or await Theorem 19. But H is not an a*-extension of G [11,Example 4.11].…”

Section: Let (G S) Have Closed Stabilizers Then G C Is An N-extensimentioning

confidence: 99%

The lateral completion of a completely distributive lattice-ordered group (revisited)

Davis

McCleary

1981

J Aust Math Soc A

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“…It was found [2] that these groups are not always /-simple. Pathological groups played a crucial role in [3] in the theory of a*-extensions and t-extensions, where the two groups G and H (defined below) arose quite naturally. (H, R) turned out to be the unique f-closure of the groups (G, R), (H, R), and (G, R).…”

mentioning

confidence: 99%

“…Let These groups were mentioned briefly in [3] in connection with f-extensions, but without proof even of the fact they are groups. Here we fill in some details, and establish their /-simplicity.…”

mentioning

confidence: 99%

Some 𝑙-simple pathological lattice-ordered groups

Glass¹,

McCleary²

1976

Proc. Amer. Math. Soc.

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Abstract.The purpose of this note is to explore two recently discovered lattice-ordered groups, and to establish that they are /-simple, i.e., have no proper /-ideals. These groups are "pathologically o-2-transitive" ordered permutation groups acting on the real line. The two groups exhibit even nastier properties than previous pathological groups, and serve as counterexamples to several conjectures about /-groups.

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a^∗-closures of completely distributive lattice-ordered groups

Cited by 7 publications

References 13 publications

Tyings in lattice-ordered permutation groups

Tyings in lattice-ordered permutation groups

The lateral completion of a completely distributive lattice-ordered group (revisited)

Some 𝑙-simple pathological lattice-ordered groups

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a∗-closures of completely distributive lattice-ordered groups

Cited by 7 publications

References 13 publications

Tyings in lattice-ordered permutation groups

Tyings in lattice-ordered permutation groups

The lateral completion of a completely distributive lattice-ordered group (revisited)

Some 𝑙-simple pathological lattice-ordered groups

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a^∗-closures of completely distributive lattice-ordered groups