convex subgroup

convex subgroup

We begin this article with something more general. Let

P

be a poset. A subset

A

⊆

P

is said to be

convex

if for any

a

,

b

∈

A

with

a

≤

b

, the

poset interval

[

a

,

b

]

⊆

A

also. In other words,

c

∈

A

for any

c

∈

P

such that

a

≤

c

and

c

≤

b

. Examples of

convex subsets

are

intervals

themselves,

antichains

, whose intervals are

singletons

, and the

empty set

.

One encounters

convex sets

most often in the study of

partially ordered groups

. A

convex subgroup

H

of a po-group

G

is a

subgroup

of

G

that is a convex subset of the poset

G

at the same time. Since

e

∈

H

, we have that

[

e

,

a

]

⊆

H

for any

e

≤

a

∈

H

.

Conversely

, if a subgroup

H

satisfies

the

property

that

[

e

,

a

]

⊆

H

whenever

a

∈

H

, then

H

is a convex subgroup: if

a

,

b

∈

H

, then

a

-

1

⁢

b

∈

H

, so that

[

e

,

a

-

1

⁢

b

]

⊆

H

, which implies that

[

a

,

b

]

=

a

⁢

[

e

,

a

-

1

⁢

b

]

⊆

H

as well.

For example, let

G

=

ℝ

2

be the po-group under the usual

Cartesian ordering

.

G

and

0

are both convex, but these are

trivial

examples. Let us see what other convex subgroups

H

there are. Suppose

P

=

(

a

,

b

)

∈

H

with

(

a

,

b

)

≠

(

0

,

0

)

=

O

. We divide this into several cases:

1.

a

⁢

b

>

0

. If

a

>

0

, then

b

>

0

(

P

in the first

quadrant

), so that

O

≤

P

, which means

[

O

,

P

]

⊆

H

. If

a

<

0

, then

b

<

0

(

P

in the third quandrant), so that

O

≤

-

P

. In either case,

H

contains a

rectangle

(

[

O

,

P

]

or

[

O

,

-

P

]

) that generates

G

, so

H

=

G

.

2.

One of

a

or

b

is

0

. Suppose

a

=

0

for now. Then either

0

<

b

so that

[

O

,

P

]

⊆

H

or

b

<

0

so that

[

O

,

-

P

]

⊆

H

. In either case,

H

contains a

line segment

on the

y

-axis. But this line segment generates the

y

-axis. So

y

-axis

⊆

H

. If

H

is a subgroup of the

y

-axis, then

H

=

y

-axis.

Otherwise, another point

Q

=

(

c

,

d

)

∈

H

not on the

y

-axis. We have the following subcases:

(a)

If

c

⁢

d

>

0

, then

H

=

G

as in the previous case.

(b)

If

c

⁢

d

<

0

, say

d

<

0

(or

0

<

c

), then for some positive integer

n

,

0

<

d

+

n

⁢

b

, so that

O

≤

Q

+

n

⁢

P

, and

H

=

G

as well. On the other hand, if

c

<

0

(or

0

<

d

), then

-

Q

returns us to the previous

argument

and

H

=

G

again.

(c)

If

d

=

0

(so

c

≠

0

), then either

O

≤

P

+

Q

(when

0

<

c

) or

O

≤

P

-

Q

(when

c

<

0

), so that

H

=

G

once more.

A similar set of arguments shows that if

H

contains a segment of the

x

-axis, then either

H

is the

x

-axis or

H

=

G

. In

conclusion

, in the case when

a

⁢

b

=

0

,

H

is either one of the two axes, or the entire group.

3.

a

⁢

b

<

0

. It is enough to assume that

0

<

a

and

b

<

0

(that

P

lies in the fourth quadrant), for if

P

lies in the second quadrant,

-

P

lies in the fourth.

Since

O

,

P

∈

H

,

H

could be a subgroup of the line group

L

containing

O

and

P

. No two points on

L

are

comparable

, for if

(

r

,

s

)

<

(

t

,

u

)

on

L

, then the slope of

L

is

positive

0

<

u

-

s

t

-

r

,

a

contradiction

. So

L

, and hence

H

, is an antichaine. This means that

H

is convex.

Suppose now

H

contains a point

Q

=

(

c

,

d

)

not on

L

. We again break this down into subcases:

(a)

Q

is in the first or third quandrant. Then

H

=

G

as in the very first case above.

(b)

Q

is on either of the axes. Then

H

=

G

also, as in case 2(b) above.

(c)

Q

is in the second or fourth quadrant. It is enough to assume that

Q

is in the same quadrant as

P

(fourth). So we have

0

<

c

and

d

<

0

. Since

L

passes through

P

and not

Q

, we have that

a

c

≠

b

d

.

Let

0

<

r

=

a

/

c

and

0

<

s

=

b

/

d

and assume

r

<

s

. Then there is a

rational number

m

/

n

(with

0

<

m

,

n

) such that

r

<

m

n

<

s

.

This means that

n

⁢

a

<

m

⁢

c

and

n

⁢

b

<

m

⁢

d

, or

n

⁢

P

<

m

⁢

Q

. But

n

⁢

P

,

m

⁢

Q

∈

H

, so is

R

=

m

⁢

Q

-

n

⁢

P

∈

H

, which is in the first quadrant. This implies that

H

=

G

too.

In summary, if

H

contains a point in the second or fourth quadrant, then either

H

is a subgroup of a line with slope

<

0

, or

H

=

G

.

The three main cases above exhaust all convex subgroups of

ℝ

2

under the Cartesian ordering.

If the

Euclidean plane

is equipped with the lexicographic ordering, then the story is quite different, but simpler. If

H

is non-trivial, say

P

=

(

a

,

b

)

∈

H

,

P

≠

O

. If

0

<

a

, then

(

c

,

d

)

≤

(

a

,

b

)

for any

c

<

a

regardless of

d

. Choose

Q

=

(

c

,

d

)

to be in the first quadrant. Then

[

O

,

Q

]

⊆

H

, so that

H

=

G

. If

a

<

0

, then

-

P

takes us back to the previous argument. If

a

=

0

, then either

[

O

,

P

]

(when

0

<

b

), or

[

O

,

-

P

]

(when

b

<

0

) is a positive interval on the

y

-axis. This implies that

H

is at least the

y

-axis. If

H

contains no other points, then

H

=

y

-axis. In summary, the po-group

ℝ

2

with

lexicographic order

has the

y

-axis as the only non-trivial proper convex subgroup.

References

1

G. Birkhoff

Lattice Theory

, 3rd Edition, AMS Volume XXV, (1967).

Title

convex subgroup

Canonical

name

ConvexSubgroup

Date of creation

2013-03-22 17:04:04

Last modified on

2013-03-22 17:04:04

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

4

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06A99

Classification

msc 06F15

Classification

msc 06F20

Classification

msc 20F60

Defines

convex subset

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