Closed subgroups of lattice-ordered permutation groups
Authors: McCleary Stephen H. 
Published: July 2010

Let

G
G

be an

l
l

-subgroup of the lattice-ordered group

A
(
Ω
)

A(\Omega )

of order-preserving permutations of a chain

Ω
\Omega

; and in this abstract, assume for convenience that

G
G

is transitive. Let

Ω
¯

\bar \Omega

denote the completion by Dedekind cuts of

Ω
\Omega

. The stabilizer subgroups

G

ω
¯

=
{
g
ϵ
G

|

ω
¯

g
=

ω
¯

}
,

ω
¯

ϵ

Ω
¯

{G_{\bar \omega }} = \{ g \epsilon G|\bar \omega g = \bar \omega \} ,\bar \omega \epsilon \bar \Omega

, will be used to characterize certain subgroups of

G
G

which are closed (under arbitrary suprema which exist in

G
G

). If

Δ
\Delta

is an

o
o

-block of

G
G

(a nonempty convex subset such that for any

g
ϵ
G

g \epsilon G

, either

Δ
g
=
Δ

\Delta g = \Delta

or

Δ
g
∩
Δ

\Delta g \cap \Delta

is empty), and if

ω
¯

=
sup
Δ
,

G
Δ

\bar \omega = \sup \Delta ,{G_\Delta }

will denote

{
g
ϵ
G

|

Δ
g
=
Δ
}
=

G

ω
¯

\{ g \epsilon G|\Delta g = \Delta \} = {G_{\bar \omega }}

; and the

o
o

-block system

Δ
~

\tilde \Delta

consisting of the translates

Δ
g

\Delta g

of

Δ
\Delta

will be called closed if

G
Δ

{G_\Delta }

is closed. When the collection of

o
o

-block systems is totally ordered (by inclusion, viewing the systems as congruences), there is a smallest closed system

C

\mathcal {C}

, and all systems above

C

\mathcal {C}

are closed.

C

\mathcal {C}

is the trivial system (of singletons) iff

G
G

is complete (in

A
(
Ω
)

A(\Omega )

).

G

ω
¯

{G_{\bar \omega }}

is closed iff

ω
¯

\bar \omega

is a cut in

C

\mathcal {C}

i.e.,

ω
¯

\bar \omega

is not in the interior of any

Δ
ϵ

C

\Delta \epsilon \mathcal {C}

. Every closed convex

l
l

-subgroup of

G
G

is an inter-section of stabilizers of cuts in

C

\mathcal {C}

. Every closed prime subgroup

≠
G

\ne G

is either a stabilizer of a cut in

C

\mathcal {C}

, or else is minimal and is the intersection of a tower of such stabilizers.

L
(

C

)
=
∩
{

G
Δ

|

Δ
ϵ

C

}

L(\mathcal {C}) = \cap \{ {G_\Delta }|\Delta \epsilon \mathcal {C}\}

is the distributive radical of

G
G

, so that

G
G

acts faithfully (and completely) on

C

\mathcal {C}

iff

G
G

is completely distributive. Every closed

l
l

-ideal of

G
G

is

L
(

D

)

L(\mathcal {D})

for some system

D

\mathcal {D}

. A group

G
G

in which every nontrivial

o
o

-block supports some

1
≠
g
ϵ
G

1 \ne g \epsilon G

(e.g., a generalized ordered wreath product) fails to be complete iff

G
G

has a smallest nontrivial system

Δ
~

\tilde \Delta

and the restriction

G
Δ

|

Δ

{G_\Delta }|\Delta

is

o
o

-

2
2

-transitive and lacks elements

≠
1

\ne 1

of bounded support. These results about permutation groups are used to show that if

H
H

is an abstract

l
l

-group having a representing subgroup, its closed

l
l

-ideals form a tower under inclusion; and that if

{

K
λ

}

\{ {K_\lambda }\}

is a Holland kernel of a completely distributive abstract

l
l

-group

H
H

, then so is the set of closures

{

K
λ
∗

}

\{ K_\lambda ^ \ast \}

, so that if

H
H

has a transitive representation as a permutation group, it has a complete transitive representation.