aarjaneiro/planetmath huggingface dataset:
lowest upper bound

lowest upper bound

Let

S

be a set with a

partial ordering

≤

, and let

T

be a subset of

S

. A

lowest upper bound

, or

supremum

, of

T

is an

upper bound

x

of

T

with the

property

that

x

≤

y

for every upper bound

y

of

T

. The lowest upper bound of

T

, when it exists, is denoted

sup

⁡

(

T

)

.

A lowest upper bound of

T

, when it exists, is unique.

Greatest lower bound

is defined similarly: a

greatest lower bound

, or

infimum

, of

T

is a

lower bound

x

of

T

with the property that

x

≥

y

for every lower bound

y

of

T

. The greatest lower bound of

T

, when it exists, is denoted

inf

⁡

(

T

)

.

If

A

=

{

a

1

,

a

2

,

…

,

a

n

}

is a

finite set

, then the supremum of

A

is simply

max

⁡

(

A

)

, and the infimum of

A

is equal to

min

⁡

(

A

)

.

Title

lowest upper bound

Canonical

name

LowestUpperBound

Date of creation

2013-03-22 11:52:18

Last modified on

2013-03-22 11:52:18

Owner

djao (24)

Last modified by

djao (24)

Numerical id

13

Author

djao (24)

Entry type

Definition

Classification

msc 06A05

Defines

least upper bound

Defines

greatest lower bound

Defines

supremum

Defines

infimum

Generated on Fri Feb 9 09:50:21 2018 by

LaTeXML
order of six means

order of six means

The of the six usual means of two positive numbers (

a

and

b

) is from the least to the greatest one

1.

harmonic mean

,

2.

geometric mean

,

3.

Heronian mean

,

4.

arithmetic mean

,

5.

quadratic mean

,

6.

contraharmonic mean

,

i. e.

2

⁢

a

⁢

b

a

+

b

≦

a

⁢

b

≦

a

+

a

⁢

b

+

b

3

≦

a

+

b

2

≦

a

2

+

b

2

2

≦

a

2

+

b

2

a

+

b

.

The equality signs are valid iff

a

=

b

.

Proof.

If

x

2

-

y

2

≧

0

for nonnegative

x

and

y

,  then

x

≧

y

.

“

1

≦

2

”:

(

a

⁢

b

)

2

-

(

a

+

b

2

)

2

=

a

⁢

b

-

4

⁢

a

2

⁢

b

2

(

a

+

b

)

2

=

a

⁢

b

⁢

(

1

-

4

⁢

a

⁢

b

(

a

+

b

)

2

)

=

a

⁢

b

⋅

(

a

+

b

)

2

-

4

⁢

a

⁢

b

(

a

+

b

)

2

=

a

⁢

b

⁢

(

a

-

b

)

2

(

a

+

b

)

2

≧

0

“

2

≦

3

” and “

3

≦

4

”: proven in

Heronian mean is between geometric and arithmetic mean

“

4

≦

5

”:

(

a

2

+

b

2

2

)

2

-

(

a

+

b

2

)

2

=

2

⁢

a

2

+

2

⁢

b

2

-

a

2

-

2

⁢

a

⁢

b

-

b

2

4

=

(

a

-

b

2

)

2

≧

0

“

5

≦

6

”:

(

a

2

+

b

2

a

+

b

)

2

-

(

a

2

+

b

2

2

)

2

=

2

⁢

(

a

2

+

b

2

)

2

-

(

a

2

+

b

2

)

⁢

(

a

+

b

)

2

2

⁢

(

a

+

b

)

2

=

(

a

2

+

b

2

)

⁢

(

2

⁢

a

2

+

2

⁢

b

2

-

a

2

-

2

⁢

a

⁢

b

-

b

2

)

2

⁢

(

a

+

b

)

2

=

(

a

2

+

b

2

)

⁢

(

a

-

b

)

2

2

⁢

(

a

+

b

)

2

≧

0

Title

order of six means

Canonical

name

OrderOfSixMeans

Date of creation

2013-03-22 18:45:28

Last modified on

2013-03-22 18:45:28

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

4

Author

pahio (2872)

Entry type

Theorem

Classification

msc 06A05

Classification

msc 26B35

Classification

msc 26D07

Related topic

Mean3

Related topic

ComparisonOfPythagoreanMeans

Related topic

InequalityWithAbsoluteValues

Related topic

LehmerMean

Generated on Fri Feb 9 09:50:24 2018 by

LaTeXML
ordered group

ordered group

Definition 1.

We say that the

subsemigroup

S

of the group

G

(with the

operation

denoted multiplicatively) defines an

G

, if

•

a

-

1

⁢

S

⁢

a

⊆

S

∀

a

∈

G

,

•

G

=

S

∪

{

1

}

∪

S

-

1

where

S

-

1

=

{

s

-

1

:

s

∈

S

}

and the members of the union are

pairwise disjoint

.

The order “

<

” of the group

G

is explicitly given by setting in

G

:

a

<

b

⇔

a

⁢

b

-

1

∈

S

Then we speak of the

ordered group

(

G

,

<

)

,  or simply

G

.

Theorem 1

.

The order “

<

” defined by the subsemigroup

S

of the group

G

has the following properties.

1.

For all

a

,

b

∈

G

, exactly one of the conditions

a

<

b

,

a

=

b

,

b

<

a

holds.

2.

a

<

b

∧

b

<

c

⇒

a

<

c

3.

a

<

b

⇒

a

⁢

c

<

b

⁢

c

∧

c

⁢

a

<

c

⁢

b

4.

a

<

b

∧

c

<

d

⇒

a

⁢

c

<

b

⁢

d

5.

a

<

b

⇔

b

-

1

<

a

-

1

6.

a

<

1

⇔

a

∈

S

Definition 2.

The set

G

is an

ordered group equipped with zero

0, if the set

G

*

of its

elements

distinct from its element 0 forms an ordered group

(

G

*

,

<

)

and if

•

0

⁢

a

=

a

⁢

0

=

0

∀

a

∈

G

,

•

0

<

a

∀

a

∈

G

*

.

Cf. 7 in

examples of semigroups

.

References

1

Emil Artin

:

Theory of

Algebraic Numbers

.  Lecture notes.  Mathematisches Institut, Göttingen (1959).

2

Paul Jaffard

:

Les systèmes d’idéaux

.  Dunod, Paris (1960).

Title

ordered group

Canonical name

OrderedGroup

Date of creation

2013-03-22 14:54:36

Last modified on

2013-03-22 14:54:36

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

16

Author

pahio (2872)

Entry type

Definition

Classification

msc 06A05

Classification

msc 20F60

Related topic

KrullValuation

Related topic

PartiallyOrderedGroup

Related topic

PraeclarumTheorema

Defines

ordered group equipped with zero

Generated on Fri Feb 9 09:50:26 2018 by

LaTeXML
supremum over closure

supremum over closure

Theorem 1

.

Let

f

:

R

→

R

be a

continuous function

and

A

⊆

R

. Then

sup

x

∈

A

⁡

f

⁢

(

x

)

=

sup

x

∈

A

¯

⁡

f

⁢

(

x

)

, where

A

¯

denotes the

closure

of

A

.

Proof.

The

theorem

is clearly true for

A

=

∅

. Thus, it will be assumed that

A

≠

∅

.

Since

A

⊆

A

¯

, we have

sup

x

∈

A

⁡

f

⁢

(

x

)

≤

sup

x

∈

A

¯

⁡

f

⁢

(

x

)

.

Suppose first that

sup

x

∈

A

¯

⁡

f

⁢

(

x

)

=

∞

. Let

r

∈

ℝ

. Then there exists

x

0

∈

A

¯

with

f

⁢

(

x

0

)

≥

r

+

1

. Since

f

is

continuous

, there exists

δ

>

0

such that, for any

x

∈

ℝ

with

-

δ

<

x

-

x

0

<

δ

, we have

-

1

<

f

⁢

(

x

)

-

f

⁢

(

x

0

)

<

1

. Since

x

0

∈

A

¯

, there exists

x

1

∈

A

with

-

δ

<

x

1

-

x

0

<

δ

. (Recall that

x

∈

A

¯

if and only if every

neighborhood

of

x

intersects

A

.) Thus,

f

⁢

(

x

1

)

-

f

⁢

(

x

0

)

>

-

1

. Therefore,

f

⁢

(

x

1

)

>

f

⁢

(

x

0

)

-

1

≥

r

+

1

-

1

=

r

. Hence,

sup

x

∈

A

⁡

f

⁢

(

x

)

=

∞

.

Now suppose that

sup

x

∈

A

¯

⁡

f

⁢

(

x

)

=

R

for some

R

∈

ℝ

. Let

ε

>

0

. Then there exists

x

2

∈

A

¯

with

f

⁢

(

x

2

)

≥

R

-

ε

2

. Since

f

is continuous, there exists

δ

′

>

0

such that, for any

x

∈

ℝ

with

-

δ

′

<

x

-

x

0

<

δ

′

, we have

-

ε

2

<

f

⁢

(

x

)

-

f

⁢

(

x

2

)

<

ε

2

. Since

x

2

∈

A

¯

, there exists

x

3

∈

A

with

-

δ

′

<

x

3

-

x

2

<

δ

′

. Thus,

f

⁢

(

x

3

)

-

f

⁢

(

x

2

)

>

-

ε

2

. Therefore,

f

⁢

(

x

3

)

>

f

⁢

(

x

2

)

-

ε

2

≥

R

-

ε

2

-

ε

2

=

R

-

ε

. Hence,

sup

x

∈

A

⁡

f

⁢

(

x

)

≥

R

.

In either case, it follows that

sup

x

∈

A

⁡

f

⁢

(

x

)

=

sup

x

∈

A

¯

⁡

f

⁢

(

x

)

.
∎

Note that this theorem also holds for continuous functions

f

:

X

→

ℝ

, where

X

is an arbitrary

topological space

. To prove this fact, one would need to slightly adjust the proof supplied here.

Title

supremum over closure

Canonical name

SupremumOverClosure

Date of creation

2013-03-22 17:08:22

Last modified on

2013-03-22 17:08:22

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

9

Author

Wkbj79 (1863)

Entry type

Theorem

Classification

msc 06A05

Classification

msc 26A15

Generated on Fri Feb 9 09:50:33 2018 by

LaTeXML
total order

total order

A

totally ordered set

(or

linearly ordered set

) is a poset

(

T

,

≤

)

which has the property of

comparability

:

•

for all

x

,

y

∈

T

, either

x

≤

y

or

y

≤

x

.

In other words, a totally ordered set is a set

T

with a

binary relation

≤

on it
such that the following hold for all

x

,

y

,

z

∈

T

:

•

x

≤

x

. (

reflexivity

)

•

If

x

≤

y

and

y

≤

x

, then

x

=

y

. (

antisymmetry

)

•

If

x

≤

y

and

y

≤

z

, then

x

≤

z

. (

transitivity

)

•

Either

x

≤

y

or

y

≤

x

. (

comparability

)

The binary relation

≤

is then called a

total order

or a

linear order

(or

total ordering

or

linear ordering

).
A totally ordered set is also sometimes called a

chain

, especially when it is considered as a subset of some other poset.
If every nonempty subset of

T

has a least

element

, then the total order is called a well-order (

http://planetmath.org/WellOrderedSet

).

Some people prefer to define the binary relation

<

as a total order, rather than

≤

.
In this case,

<

is required to be

transitive

(

http://planetmath.org/Transitive3

) and to obey the

law of trichotomy

.
It is straightforward to check that this is

equivalent

to the above definition, with the usual relationship between

<

and

≤

(that is,

x

≤

y

if and only if either

x

<

y

or

x

=

y

).

A totally ordered set can also be defined as a

lattice

(

T

,

∨

,

∧

)

in which the following property holds:

•

for all

x

,

y

∈

T

, either

x

∧

y

=

x

or

x

∧

y

=

y

.

Then totally ordered sets are

distributive

lattices (

http://planetmath.org/DistributiveLattice

).

Title

total order

Canonical name

TotalOrder

Date of creation

2013-03-22 11:43:35

Last modified on

2013-03-22 11:43:35

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

25

Author

yark (2760)

Entry type

Definition

Classification

msc 06A05

Classification

msc 91B12

Classification

msc 55-00

Classification

msc 55-01

Synonym

linear order

Synonym

total ordering

Synonym

linear ordering

Related topic

PartialOrder

Related topic

Relation

Related topic

SortingProblem

Related topic

OrderedRing

Related topic

ProofOfGeneralizedIntermediateValueTheorem

Related topic

LinearContinuum

Defines

totally ordered set

Defines

linearly ordered set

Defines

comparability

Defines

totally ordered

Defines

linearly ordered

Defines

chain

Defines

totally-ordered set

Defines

linearly-ordered set

Defines

totally-ordered

Defines

linearly-ordered

Generated on Fri Feb 9 09:50:35 2018 by

LaTeXML
atom

atom

Let

P

be a poset, partially ordered by

≤

. An

element

a

∈

P

is called an

atom

if it covers some

minimal element

of

P

. As a result, an atom is never

minimal

. A poset

P

is called

atomic

if for every element

p

∈

P

that is not minimal has an atom

a

such that

a

≤

p

.

Examples

.

1.

Let

A

be a set and

P

=

2

A

its

power set

.

P

is a poset ordered by

⊆

with a unique minimal element

∅

. Thus, all singleton subsets

{

a

}

of

A

are atoms in

P

.

2.

ℤ

+

is partially ordered if we define

a

≤

b

to mean that

a

∣

b

. Then

1

is a minimal element and any prime number

p

is an atom.

Remark.

Given a

lattice

L

with underlying poset

P

, an element

a

∈

L

is called an

atom

(of

L

) if it is an atom in

P

. A lattice is a called an

atomic lattice

if its underlying poset is atomic. An

atomistic lattice

is an atomic lattice such that each element that is not minimal is a join of atoms. If

a

is an atom in a

semimodular lattice

L

, and if

a

is not under

x

, then

a

∨

x

is an atom in any

interval

lattice

I

where

x

=

⋀

I

.

Examples

.

1.

P

=

2

A

, with the usual

intersection

and union as the lattice

operations

meet and join, is atomistic: every subset

B

of

A

is the union of all the singleton subsets of

B

.

2.

ℤ

+

, partially ordered as above, with lattice

binary operations

defined by

a

∧

b

=

gcd

⁡

(

a

,

b

)

, and

a

∨

b

=

lcm

⁡

(

a

,

b

)

, is a lattice that is atomic, as we have seen earlier. But it is not atomistic:

4

is not a join of

2

’s;

36

is not a join of

2

and

3

are just two

counterexamples

.

Title

atom

Canonical name

Atom

Date of creation

2013-03-22 15:20:09

Last modified on

2013-03-22 15:20:09

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

13

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06A06

Classification

msc 06B99

Defines

atomic poset

Defines

atomic lattice

Defines

atomistic lattice

Defines

atomistic

Generated on Fri Feb 9 09:50:45 2018 by

LaTeXML
chain finite

chain finite

A poset is said to be

chain finite

if every chain with both
maximal (

http://planetmath.org/MaximalElement

) and

minimal element

is finite.

ℤ

with the standard order relation is chain finite,
since any

infinite subset

of

ℤ

must be

unbounded

(

http://planetmath.org/UpperBound

) above or below.

ℚ

with the standard order relation is not chain finite,
since for example

{

x

∈

ℚ

∣

0

⩽

x

⩽

1

}

is

infinite

and has both a

maximal element

1

and a minimal element

0

.

Chain finiteness is often used to draw

conclusions

about an order from information about its covering relation (or equivalently, from its

Hasse diagram

).

Title

chain finite

Canonical

name

ChainFinite

Date of creation

2013-03-22 16:55:07

Last modified on

2013-03-22 16:55:07

Owner

lars_h (9802)

Last modified by

lars_h (9802)

Numerical id

4

Author

lars_h (9802)

Entry type

Definition

Classification

msc 06A06

Defines

chain finite

Generated on Fri Feb 9 09:50:47 2018 by

LaTeXML
Dilworth’s theorem

Dilworth’s theorem

Theorem

.

If

P

is a poset with width

w

<

∞

, then

w

is also the smallest integer such that

P

can be written as the union of

w

chains.

Remark

. The smallest

cardinal

c

such that

P

can be written as the union of

c

chains is called the

chain covering number

of

P

. So Dilworth’s

theorem

says that if the width of

P

is finite, then it is equal to the chain covering number of

P

. If

w

is

infinite

, then statement is not true. The proof of Dilworth’s theorem and its

counterexample

in the infinite case can be found in the reference below.

References

1

J.B. Nation, “

Lattice Theory

”,

http://www.math.hawaii.edu/ jb/lat1-6.pdf

http://www.math.hawaii.edu/ jb/lat1-6.pdf

Title

Dilworth’s theorem

Canonical

name

DilworthsTheorem

Date of creation

2013-03-22 15:49:37

Last modified on

2013-03-22 15:49:37

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

14

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 06A06

Classification

msc 06A07

Synonym

Dilworth chain decomposition theorem

Related topic

DualOfDilworthsTheorem

Defines

chain covering number

Generated on Fri Feb 9 09:50:49 2018 by

LaTeXML
dimension of a poset

dimension of a poset

Let

P

be a finite poset and

ℛ

be the family of all

realizers

of

P

. The

dimension

of

P

, written

dim

⁡

(

P

)

, is the

cardinality

of a member

E

∈

ℛ

with the smallest cardinality. In other words, the dimension

n

of

P

is the least number of

linear extensions

L

1

,

…

,

L

n

of

P

such that

P

=

L

1

∩

⋯

∩

L

n

. (

E

can be chosen to be

{

L

1

,

…

,

L

n

}

).

If

P

is a chain, then

dim

⁡

(

P

)

=

1

. The

converse

is clearly true too. An example of a poset with dimension 2 is an

antichain

with at least

2

elements

. For if

P

=

{

a

1

,

…

,

a

m

}

is an antichain, then one way to linearly extend

P

is to simply put

a

i

≤

a

j

iff

i

≤

j

. Called this

extension

L

1

. Another way to order

P

is to reverse

L

1

, by

a

i

≤

a

j

iff

j

≤

i

. Call this

L

2

. Note that

L

1

and

L

2

are duals of each other. Let

L

=

L

1

∩

L

2

. As both

L

1

and

L

2

are linear extensions of

P

,

P

⊆

L

. On the other hand, if

(

a

i

,

a

j

)

∈

L

, then

a

i

≤

a

j

in both

L

1

and

L

2

, so that

i

≤

j

and

j

≤

i

, or

i

=

j

and whence

a

i

=

a

j

, which implies

(

a

i

,

a

j

)

=

(

a

i

,

a

i

)

∈

P

.

L

⊆

P

and thus

dim

⁡

(

P

)

=

2

.

Remark

. Let

P

be a finite poset. A

theorem

of Dushnik and Miller states that the smallest

n

such that

P

can be embedded in

ℝ

n

, considered as the

n

-fold

product of posets

, or chains of real numbers

ℝ

, is the dimension of

P

.

References

1

W. T. Trotter,

Combinatorics

and

Partially Ordered Sets

, Johns-Hopkins University Press, Baltimore (1992).

Title

dimension of a poset

Canonical

name

DimensionOfAPoset

Date of creation

2013-03-22 16:33:29

Last modified on

2013-03-22 16:33:29

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06A06

Classification

msc 06A07

Defines

dimension

Generated on Fri Feb 9 09:50:58 2018 by

LaTeXML
directed set

directed set

A

directed set

is a

partially ordered set

(

A

,

≤

)

such that whenever

a

,

b

∈

A

there is an

x

∈

A

such that

a

≤

x

and

b

≤

x

.

A subset

B

⊆

A

is said to be

residual

if there is

a

∈

A

such that

b

∈

B

whenever

a

≤

b

, and

cofinal

if for each

a

∈

A

there is

b

∈

B

such that

a

≤

b

.

A directed set is sometimes called an

upward-directed set

.
We may also define the dual notion:
a

downward-directed set

(or

filtered set

) is a partially ordered set

(

A

,

≤

)

such that whenever

a

,

b

∈

A

there is an

x

∈

A

such that

x

≤

a

and

x

≤

b

.

Note: Many authors do not require

≤

to be antisymmetric,
so that it is only a

pre-order

(rather than a

partial order

)
with the given

property

.
Also, it is common to require

A

to be non-empty.

Title

directed set

Canonical

name

DirectedSet

Date of creation

2013-03-22 12:54:00

Last modified on

2013-03-22 12:54:00

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

11

Author

yark (2760)

Entry type

Definition

Classification

msc 06A06

Synonym

upward-directed set

Synonym

upward directed set

Related topic

Cofinality

Related topic

AccumulationPointsAndConvergentSubnets

Defines

residual

Defines

cofinal

Defines

downward-directed set

Defines

downward directed set

Defines

filtered set

Generated on Fri Feb 9 09:51:00 2018 by

LaTeXML
disjunction property of Wallman

disjunction property of Wallman

A

partially ordered set

𝔄

with a least

element

0

has the

disjunction property of Wallman

if for every pair

(

a

,

b

)

of elements of the poset, either

b

≤

a

or there exists an element

c

≤

b

such that

c

≠

0

and

c

has no nontrivial common

predecessor

with

a

. That is, in the latter case, the only

x

with

x

≤

a

and

x

≤

c

is

x

=

0

.

For the case if the poset

𝔄

is a

∩

-

semilattice

disjunction property of Wallman

is

equivalent

to every of the following three

formulas

:

1.

∀

a

,

b

∈

𝔄

:

(

{

c

∈

𝔄

|

c

∩

a

≠

0

}

=

{

c

∈

𝔄

|

c

∩

b

≠

0

}

⇒

a

=

b

)

;

2.

∀

a

,

b

∈

𝔄

:

(

{

c

∈

𝔄

|

c

∩

a

≠

0

}

⊆

{

c

∈

𝔄

|

c

∩

b

≠

0

}

⇒

a

⊆

b

)

;

3.

∀

a

,

b

∈

𝔄

:

(

a

⊂

b

⇒

{

c

∈

𝔄

|

c

∩

a

≠

0

}

⊂

{

c

∈

𝔄

|

c

∩

b

≠

0

}

)

.

The proof of this equivalence can be found in

http://www.mathematics21.org/binaries/filters.pdf

this online article.

Title

disjunction property of Wallman

Canonical name

DisjunctionPropertyOfWallman

Date of creation

2013-03-22 17:53:48

Last modified on

2013-03-22 17:53:48

Owner

porton (9363)

Last modified by

porton (9363)

Numerical id

7

Author

porton (9363)

Entry type

Definition

Classification

msc 06A06

Synonym

Wallman’s

disjunction property

Related topic

Poset

Generated on Fri Feb 9 09:51:02 2018 by

LaTeXML
dual of Dilworth’s theorem

dual of Dilworth’s theorem

Theorem 1

.

Let

P

be a poset of height

h

. Then

P

can be partitioned into

h

antichains

and furthermore at least

h

antichains are required.

Proof.

Induction

on

h

. If

h

=

1

, then no elements of

P

are

comparable

, so

P

is an antichain. Now suppose that

P

has height

h

≥

2

and that the

theorem

is true for

h

-

1

. Let

A

1

be the set of

maximal elements

of

P

. Then

A

1

is an antichain in

P

and

P

-

A

1

has height

h

-

1

since we have removed precisely one

element

from every chain. Hence,

P

-

A

1

can be paritioned into

h

-

1

antichains

A

2

,

A

3

,

…

⁢

A

h

. Now we have the

partition

A

1

∪

A

2

∪

…

∪

A

h

of

P

into

h

antichains as desired.

The

necessity

of

h

antichains is

trivial

by the

pigeonhole principle

; since

P

has height

h

, it has a chain of length

h

, and each element of this chain must be placed in a different antichain of our partition.
∎

Title

dual of Dilworth’s theorem

Canonical name

DualOfDilworthsTheorem

Date of creation

2013-03-22 15:01:49

Last modified on

2013-03-22 15:01:49

Owner

justice (4961)

Last modified by

justice (4961)

Numerical id

7

Author

justice (4961)

Entry type

Theorem

Classification

msc 06A06

Related topic

DilworthsTheorem

Generated on Fri Feb 9 09:51:04 2018 by

LaTeXML
flag

flag

Let

V

be a

finite-dimensional

vector space

. A

filtration

of

subspaces

V

1

⊂

V

2

⊂

⋯

⊂

V

n

=

V

is called a

flag

in

V

.
We speak of a

complete flag

when

dim

⁡

V

i

=

i

for each

i

=

1

,

…

,

n

.

Next, putting

d

k

=

dim

⁡

V

k

,

k

=

1

,

…

⁢

n

,

we say that a list of vectors

(

u

1

,

…

,

u

d

n

)

is an

adapted basis

relative to the flag, if
the first

d

1

vectors give a basis of

V

1

, the first

d

2

vectors
give a basis of

V

2

, etc. Thus, an alternate

characterization

of a
complete flag, is that the first

k

elements of an adapted basis are
a basis of

V

k

.

Example

Let us consider

ℝ

n

. For each

k

=

1

,

…

,

n

let

V

k

be the
span of

e

1

,

…

,

e

k

, where

e

j

denotes the

j

th

basic
vector, i.e. the

column vector

with

1

in the

j

th

position and
zeros everywhere else. The

V

k

give a complete flag in

ℝ

n

.
The list

(

e

1

,

e

2

,

…

,

e

n

)

is an adapted basis relative to this
flag, but the list

(

e

2

,

e

1

,

…

,

e

n

)

is not.

Generalizations.

More generally, a flag can be defined as a maximal chain in a

partially ordered set

. If one considers the poset consisting of subspaces of a (finite dimensional) vector space, one recovers the definition given above.

Title

flag

Canonical name

Flag

Date of creation

2013-03-22 12:42:35

Last modified on

2013-03-22 12:42:35

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

9

Author

rmilson (146)

Entry type

Definition

Classification

msc 06A06

Classification

msc 15A03

Defines

adapted basis

Defines

complete flag

Generated on Fri Feb 9 09:51:51 2018 by

LaTeXML