aarjaneiro/planetmath huggingface dataset:
Quasi-order is not defined uniformly

Quasi-order is not defined uniformly

In the literature, some authors define “quasi order” as

transitive

and

reflexive

, others define it as transitive and

irreflexive

.

No such discrepancy seems to exist in using “

preorder

” for the former (transitive and reflexive) and “strict

partial order

” for the latter (transitive and irreflexive).

It seems wise to use only the unambiguous terminology, and start any text where the term “quasi order” is felt with a proper warning.

Just for completeness: a

partial order

is transitive, reflexive and

antisymmetric

.

Title

Quasi-order is not defined uniformly

Canonical name

QuasiorderIsNotDefinedUniformly

Date of creation

2013-03-22 15:35:18

Last modified on

2013-03-22 15:35:18

Owner

boute (11676)

Last modified by

boute (11676)

Numerical id

5

Author

boute (11676)

Entry type

Definition

Classification

msc 06A99

Generated on Fri Feb 9 09:54:57 2018 by

LaTeXML
well quasi ordering

well quasi ordering

Let

Q

be a set and

≾

a

quasi-order

on

Q

. An

infinite

sequence

in

Q

is referred to as

bad

if for
all

i

<

j

∈

𝐍

,

a

i

≾̸

a

j

holds; otherwise it is called

good

. Note that an

antichain

is obviously a bad sequence.

A quasi-ordering

≾

on

Q

is a

well-quasi-ordering

(

wqo

) if for every infinite sequence is good. Every

well-ordering

is a well-quasi-ordering.

The following

proposition

gives

equivalent

definitions for
well-quasi-ordering:

Proposition 1

.

Given a set

Q

and a

binary relation

≾

over

Q

, the
following conditions are equivalent:

•

(

Q

,

≾

)

is a well-quasi-ordering;

•

(

Q

,

≾

)

has no infinite (

ω

-) strictly decreasing chains
and no infinite antichains.

•

Every

linear extension

of

Q

/

≈

is a well-order, where

≈

is the

equivalence relation

and

Q

/

≈

is the set of

equivalence classes

induced by

≈

.

•

Any infinite (

ω

-)

Q

-sequence contains an
increasing chain.

The equivalence of WQO to the second and the fourth conditions is proved by the infinite version of Ramsey’s

theorem

.

Title

well quasi

ordering

Canonical name

WellQuasiOrdering

Date of creation

2013-03-22 13:54:15

Last modified on

2013-03-22 13:54:15

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

13

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06A99

Classification

msc 06A07

Synonym

well quasi order

Synonym

wqo

Related topic

quasiOrder

Related topic

QuasiOrder

Generated on Fri Feb 9 09:55:10 2018 by

LaTeXML
bounded lattice

bounded lattice

A

lattice

L

is said to be

if there is an

element

0

∈

L

such that

0

≤

x

for all

x

∈

L

. Dually,

L

is

if there exists an element

1

∈

L

such that

x

≤

1

for all

x

∈

L

. A

bounded lattice

is one that is both from above and below.

For example, any finite lattice

L

is

bounded

, as

⋁

L

and

⋀

L

, being join and meet of finitely many elements, exist.

⋁

L

=

1

and

⋀

L

=

0

.

Remarks

.
Let

L

be a bounded lattice with

0

and

1

as described above.

•

0

∧

x

=

0

and

0

∨

x

=

x

for all

x

∈

L

.

•

1

∧

x

=

x

and

1

∨

x

=

1

for all

x

∈

L

.

•

As a result,

0

and

1

, if they exist, are necessarily unique. For
if there is another such a pair

0

′

and

1

′

, then

0

=

0

∧

0

′

=

0

′

∧

0

=

0

′

. Similarly

1

=

1

′

.

•

0

is called the

bottom

of

L

and

1

is called the

top

of

L

.

•

L

is a

lattice interval

and can be written as

[

0

,

1

]

.

Remark

. More generally, a poset

P

is said to be

bounded

if it has both a

greatest element

1

and a least element

0

.

Title

bounded lattice

Canonical name

BoundedLattice

Date of creation

2013-03-22 15:02:28

Last modified on

2013-03-22 15:02:28

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

13

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06B05

Classification

msc 06A06

Defines

top

Defines

bottom

Defines

bounded poset

Generated on Fri Feb 9 09:55:12 2018 by

LaTeXML
center of a lattice

center of a lattice

Let

L

be a

bounded lattice

. An

element

a

∈

L

is said to be

central

if

a

is

complemented

(

http://planetmath.org/ComplementedLattice

) and neutral (

http://planetmath.org/SpecialElementsInALattice

). The

center

of

L

, denoted

Cen

⁡

(

L

)

, is the set of all

central elements

of

L

.

Remarks

.

•

0

and

1

are central: they are

complements

of one another, both

distributive

and

dually distributive

, and satisfying the property

a

∧

b

=

a

∧

c

⁢

and

⁢

a

∨

b

=

a

∨

c

⁢

imply

⁢

b

=

c

⁢

for all

⁢

b

,

c

∈

L

where

a

∈

{

0

,

1

}

, and therefore neutral.

•

Cen

⁡

(

L

)

is a

sublattice

of

L

.

•

Cen

⁡

(

L

)

is a

Boolean algebra

.

References

1

G. Grätzer,

General

Lattice Theory

, 2nd Edition, Birkhäuser (1998).

Title

center of a

lattice

Canonical name

CenterOfALattice

Date of creation

2013-03-22 17:31:50

Last modified on

2013-03-22 17:31:50

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06B05

Defines

central element

Generated on Fri Feb 9 09:55:22 2018 by

LaTeXML
example of non-complete lattice homomorphism

example of non-complete lattice homomorphism

The real number line

[

-

∞

,

∞

]

=

ℝ

∪

{

-

∞

,

∞

}

is

complete

in its usual

ordering

of numbers. Furthermore, the meet of a subset

S

of

ℝ

is
the

infimum

of the set

S

.

Now define the map

f

:

[

-

∞

,

∞

]

→

[

-

∞

,

∞

]

as

f

⁢

(

x

)

=

{

0

x

≤

0

1

x

>

0

.

First notice that if

x

≤

y

then

f

⁢

(

x

)

≤

f

⁢

(

y

)

, for either

x

≤

y

≤

0

in
which case

f

⁢

(

x

)

=

0

=

f

⁢

(

y

)

, or

x

≤

0

<

y

which gives

f

⁢

(

x

)

=

0

<

1

=

f

⁢

(

y

)

or

0

<

x

≤

y

so

f

⁢

(

x

)

=

1

=

f

⁢

(

y

)

.

In the second place, if

S

is a finite subset of

ℝ

then

S

contains
a minimum

element

s

∈

S

. So

f

⁢

(

s

)

∈

f

⁢

(

S

)

and

f

⁢

(

s

)

≤

f

⁢

(

t

)

for all

t

∈

S

,
so

f

⁢

(

min

⁡

S

)

=

f

⁢

(

s

)

=

min

⁡

f

⁢

(

S

)

. Hence

f

is a

lattice homomorphism

.

However,

f

is not a

complete lattice homomorphism

. To see this let

S

=

{

x

∈

ℝ

:

0

<

x

}

. Then

inf

⁡

S

=

0

. However,

f

⁢

(

inf

⁡

S

)

=

f

⁢

(

0

)

=

0

while

inf

⁡

f

⁢

(

S

)

=

inf

⁡

{

1

}

=

1

.

Title

example of non-complete lattice homomorphism

Canonical name

ExampleOfNoncompleteLatticeHomomorphism

Date of creation

2013-03-22 16:58:36

Last modified on

2013-03-22 16:58:36

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

4

Author

Algeboy (12884)

Entry type

Example

Classification

msc 06B05

Classification

msc 06B99

Related topic

ExtendedRealNumbers

Generated on Fri Feb 9 09:55:24 2018 by

LaTeXML
irredundant

irredundant

Definition

. Let

L

be a

lattice

. A finite join

a

1

∨

a

2

∨

⋯

∨

a

n

of

elements

in

L

is said to be

irredundant

if one can not delete an element from from the join without resulting in a smaller join. In other words,

⋁

{

a

j

∣

j

≠

i

}

<

a

1

∨

a

2

∨

⋯

∨

a

n

for all

i

=

1

,

…

,

n

.

If the join is not irredundant, it is

redundant

Irredundant meets

are dually defined.

Remark

. The definitions above can be extended to the case where the join (or meet) is taken over an

infinite

number of elements, provided that the join (or meet) exists.

Example

. In the lattice of all subsets (ordered by

inclusion

) of

ℤ

, the set of all integers, the join

ℤ

=

⋁

{

p

⁢

ℤ

∣

p

⁢

is prime

}

is irredudant. Another irredundant join representation of

ℤ

is just the join of all atoms, the

singletons

consisting of the

individual

elements of

ℤ

. However,

ℤ

=

⋁

{

n

⁢

ℤ

∣

n

⁢

is any positive integer

}

is redundant, since

n

⁢

ℤ

can be removed whenever

n

is a

composite number

. The join of all doubletons is also redundant, for

{

a

,

b

}

≤

{

a

,

c

}

∨

{

c

,

b

}

, for any

c

∉

{

a

,

b

}

.

Definition

. An element in a lattice is

join irredundant

if it can not be written as a redundant join of elements. Dually, an element is

meet irredundant

if each of its representation as a meet of elements is irredundant.

Example

. In the two lattice diagrams (

Hasse diagram

) below,

\xymatrix

⁢

&

⁢

1

⁢

\ar

⁢

@

-

[

l

⁢

d

]

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

\ar

⁢

@

-

[

d

]

⁢

a

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

&

⁢

b

⁢

\ar

⁢

@

-

[

d

]

⁢

&

⁢

c

⁢

\ar

⁢

@

-

[

l

⁢

d

]

⁢

&

⁢

0

\xymatrix

⁢

&

⁢

1

⁢

\ar

⁢

@

-

[

l

⁢

d

]

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

a

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

&

⁢

&

⁢

b

⁢

\ar

⁢

@

-

[

l

⁢

d

]

⁢

&

⁢

0

The

1

on the left diagram is not join irredundant, since

1

=

a

∨

b

∨

c

=

a

∨

b

. On the other hand, the

1

on the right is join irredundant. Similarly, the

0

on the right is not meet irredundant, while the corresponding one on the right is.

Title

irredundant

Canonical

name

Irredundant

Date of creation

2013-03-22 18:10:08

Last modified on

2013-03-22 18:10:08

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06B05

Defines

redundant

Defines

irredundant meet

Defines

irredundant join

Defines

meet irredundant

Defines

join irredundant

Generated on Fri Feb 9 09:55:35 2018 by

LaTeXML
uniquely complemented lattice

uniquely complemented lattice

Recall that in a

bounded

distributive lattice

,

complements

,

relative complements

, and

differences of lattice elements

, if exist, must be unique. This leads to the general consideration of general bounded lattices in which complements are unique.

Definition

. A

complemented lattice

such that every

element

has a unique complement is said to be

uniquely complemented

. If

a

is an element of a

uniquely complemented lattice

,

a

′

denotes its (unique) complement. One can think of

′

as a unary

operator

on the

lattice

.

One of the first

consequences

is

a

′′

=

a

.

To see this, we have that

a

∨

a

′

=

1

,

a

∧

a

′

=

0

, as well as

a

′′

∨

a

′

=

1

,

a

′′

∧

a

′

=

0

. So

a

=

a

′′

, since they are both complements of

a

′

.

Below are some additional (and non-trivial) properties of a uniquely complemented lattice:

•

there exists a uniquely complemented lattice that is not

distributive

•

a uniquely complemented lattice

L

is distributive if at least one of the following is satisfied:

(a)

′

, as an operator on

L

, is

order reversing

;

(b)

(

a

∨

b

)

′

=

a

′

∧

b

′

;

(c)

(

a

∧

b

)

′

=

a

′

∨

b

′

;

(d)

(von Neumann)

L

is a

modular lattice

;

(e)

(Birkhoff-Ward)

L

is an

atomic lattice

.

In fact, the first three conditions are

equivalent

, so that

L

is distributive if it

satisfies

the de Morgan’s laws.

•

(Dilworth) every lattice can be embedded in a uniquely complemented lattice.

References

1

T.S. Blyth,

Lattices and Ordered

Algebraic Structures

, Springer, New York (2005).

2

G. Grätzer,

General

Lattice Theory

, 2nd Edition, Birkhäuser (1998)

Title

uniquely complemented lattice

Canonical name

UniquelyComplementedLattice

Date of creation

2013-03-22 17:58:15

Last modified on

2013-03-22 17:58:15

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06B05

Classification

msc 06C15

Defines

uniquely complemented

Generated on Fri Feb 9 09:55:47 2018 by

LaTeXML
order in an algebra

order in an algebra

Let

A

be an

algebra

(not necessarily

commutative

),

finitely generated

over

ℚ

. An

order

R

of

A

is a

subring

of

A

which is finitely generated as a

ℤ

-module and which satisfies

R

⊗

ℚ

=

A

.

Examples:

1.

The

ring of integers

in a

number field

is an order, known as
the

maximal order

.

2.

Let

K

be a

quadratic imaginary field

and

𝒪

K

its
ring of integers. For each

integer

n

≥

1

the ring

𝒪

=

ℤ

+

n

⁢

𝒪

K

is an order of

K

(in fact it can be
proved that every order of

K

is of this form). The number

n

is called the of the order

𝒪

.

Reference

: Joseph H. Silverman,

The

arithmetic of
elliptic curves

, Springer-Verlag, New York, 1986.

Title

order in an algebra

Canonical name

OrderInAnAlgebra

Date of creation

2013-03-22 13:41:22

Last modified on

2013-03-22 13:41:22

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

10

Author

alozano (2414)

Entry type

Definition

Classification

msc 06B10

Related topic

ComplexMultiplication

Defines

order

Defines

maximal order

Defines

conductor

of an order

Generated on Fri Feb 9 09:55:54 2018 by

LaTeXML
partition lattice

partition lattice

The

partition lattice

(or

lattice of partitions

)

Π

n

is the

lattice of set

partitions

(

http://planetmath.org/Partition

) of the set

[

n

]

=

{

1

,

…

,

n

}

. The

partial order

on

Π

n

is defined by

refinement

, setting

x

≤

y

if any only if each cell of

x

is

contained

in a cell of

y

.

If

n

<

3

, then

Π

n

is a chain. But

Π

3

is not even a

distributive lattice

:

Generated on Fri Feb 9 09:56:12 2018 by

LaTeXML
partitions form a lattice

partitions form a lattice

Let

S

be a set. Let

Part

⁡

(

S

)

be the set of all

partitions

on

S

. Since each partition is a cover of

S

,

Part

⁡

(

S

)

is partially ordered by

covering

refinement

relation

, so that

P

1

⪯

P

2

if for every

a

∈

P

1

, there is a

b

∈

P

2

such that

a

⊆

b

. We say that a partition

P

is finer than a partition

Q

if

P

⪯

Q

, and

coarser

than

Q

if

Q

⪯

P

.

Proposition 1

.

Part

⁡

(

S

)

is a

complete lattice

Proof.

For any set

𝒫

of partitions

P

i

of

S

, the

intersection

⋂

𝒫

is a partition of

S

. Take the meet of

P

i

to be this intersection. Next, let

𝒬

be the set of those partitions of

S

such that each

Q

∈

𝒬

is coarser than each

P

i

. This set is non-empty because

S

×

S

∈

𝒬

. Take the meet

P

′

of all these partitions which is again coarser than all partitions

P

i

. Define the join of

P

i

to be

P

′

and the proof is

complete

.
∎

Remarks

.

•

The top element of

Part

⁡

(

S

)

is

S

×

S

and the bottom is the

diagonal relation

on

S

.

•

Correspondingly, the

partition lattice

of

S

also defines the

lattice of equivalence relations

Δ

on

S

.

•

Given a family

{

E

i

∣

i

∈

I

}

of equvialence relations on

S

, we can explicitly describe the join

E

:=

⋁

E

i

of

E

i

, as follows:

a

≡

b

(

mod

E

)

iff there is a

finite sequence

a

=

c

1

,

…

,

c

n

=

b

such that

c

k

≡

c

k

+

1

(

mod

E

i

⁢

(

k

)

)

for

⁢

k

=

1

,

…

,

n

-

1

.

(1)

It is easy to see this definition makes

E

an

equivalence relation

. To see that

E

is the

supremum

of the

E

i

, first note that each

E

i

≤

E

. Suppose now

F

is an equivalence relation on

S

such that

E

i

≤

F

and

a

≡

b

(

mod

E

)

. Then we get a finite sequence

c

k

as described by (1) above, so

c

k

≡

c

k

+

1

(

mod

F

)

for each

k

∈

{

1

,

…

,

n

-

1

}

. Hence

a

≡

b

(

mod

F

)

also.

Title

partitions form a

lattice

Canonical name

PartitionsFormALattice

Date of creation

2013-03-22 16:45:22

Last modified on

2013-03-22 16:45:22

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Derivation

Classification

msc 06B20

Defines

lattice of equivalence relations

Generated on Fri Feb 9 09:56:22 2018 by

LaTeXML
closed sublattice

closed sublattice

A

sublattice

K

of a

complete lattice

L

is a

closed sublattice

of

L

iff

K

contains the meet and the join of any its nonempty subset.

Examples:

Any

complete

sublattice is a closed sublattice.

[

0

;

1

]

is a closed sublattice of

(

-

∞

;

∞

)

.

The set of rational numbers is not a closed sublattice of the set of real numbers.

Title

closed sublattice

Canonical name

ClosedSublattice

Date of creation

2013-03-22 18:57:28

Last modified on

2013-03-22 18:57:28

Owner

porton (9363)

Last modified by

porton (9363)

Numerical id

5

Author

porton (9363)

Entry type

Definition

Classification

msc 06B23

Related topic

Sublattice

Related topic

Lattice

Generated on Fri Feb 9 09:56:26 2018 by

LaTeXML
complete lattice

complete lattice

Complete lattices

A

complete lattice

is a poset

P

such that every subset of

P

has both a

supremum

and an

infimum

in

P

.

For a complete lattice

L

,
the supremum of

L

is denoted by

1

,
and the infimum of

L

is denoted by

0

.
Thus

L

is a

bounded lattice

,
with

1

as its

greatest element

and

0

as its least

element

.
Moreover,

1

is the infimum of the

empty set

,
and

0

is the supremum of the empty set.

Generalizations

A

countably complete lattice

is a poset

P

such that every

countable

subset of

P

has both a supremum and an infimum in

P

.

Let

κ

be an

infinite

cardinal

.
A

κ

-complete lattice is a

lattice

L

such that for every subset

A

⊆

L

with

|

A

|

≤

κ

, both

⋁

A

and

⋀

A

exist.
(Note that an

ℵ

0

-complete lattice
is the same as a countably complete lattice.)

Every complete lattice is a
for every infinite cardinal

κ

,
and in particular is a countably complete lattice.
Every countably complete lattice is a bounded lattice.

Title

complete lattice

Canonical name

CompleteLattice

Date of creation

2013-03-22 12:56:44

Last modified on

2013-03-22 12:56:44

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

10

Author

yark (2760)

Entry type

Definition

Classification

msc 06B23

Classification

msc 03G10

Related topic

TarskiKnasterTheorem

Related topic

CompleteLatticeHomomorphism

Related topic

Domain6

Related topic

CompleteSemilattice

Related topic

InfiniteAssociativityOfSupremumAndInfimumRegardingItself

Related topic

CompleteBooleanAlgebra

Related topic

ArbitraryJoin

Defines

countably complete lattice

Defines

countably-complete lattice

Defines

κ

-

complete

Defines

κ

-complete lattice

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complete lattice homomorphism

complete lattice homomorphism

Complete lattice homomorphism

is a

function

from one

lattice

to an other lattice, which preserves arbitrary (not only finite) meets and joins.

If

ϕ

:

L

→

M

is

lattice homomorphism

between

complete

lattices

L

and

M

such that

•

ϕ

⁢

(

⋁

{

a

i

∣

i

∈

I

}

)

=

⋁

{

ϕ

⁢

(

a

i

)

∣

i

∈

I

}

, and

•

ϕ

⁢

(

⋀

{

a

i

∣

i

∈

I

}

)

=

⋀

{

ϕ

⁢

(

a

i

)

∣

i

∈

I

}

,

then

ϕ

is called a

complete lattice homomorphism

.

Most often are considered

complete lattice homomorphisms

from one

complete lattice

to an other complete lattice (that is when all meets and joins are defined).

Complete lattice homomorphism

is a special case of lattice homomorphism.

Title

complete lattice homomorphism

Canonical name

CompleteLatticeHomomorphism

Date of creation

2013-03-22 16:58:02

Last modified on

2013-03-22 16:58:02

Owner

porton (9363)

Last modified by

porton (9363)

Numerical id

6

Author

porton (9363)

Entry type

Definition

Classification

msc 06B23

Related topic

CompleteLattice

Generated on Fri Feb 9 09:56:32 2018 by

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criteria for a poset to be a complete lattice

criteria for a poset to be a complete lattice

Proposition

. Let

L

be a poset. Then the following are

equivalent

.

1.

L

is a

complete lattice

.

2.

for every subset

A

of

L

,

⋁

A

exists.

3.

for every finite subset

F

of

L

and every

directed set

D

of

L

,

⋁

F

and

⋁

D

exist.

Proof.

Implications

1

.

⇒

2

.

⇒

3

.

are clear. We will show

3

.

⇒

2

.

⇒

1

.

(

3

.

⇒

2

.

)

If

A

=

∅

, then

⋁

A

=

0

by

definition

. So assume

A

be a non-empty subset of

L

. Let

A

′

be the set of all finite subsets of

A

and

B

=

{

⋁

F

∣

F

∈

A

′

}

. By

assumption

,

B

is

well-defined

and

A

⊆

B

. Next, let

B

′

be the set of all directed subsets of

B

, and

C

=

{

⋁

D

∣

D

∈

B

′

}

. By assumption again,

C

is well-defined and

B

⊆

C

. Now, every chain in

C

has a

maximal element

in

C

(since a chain is a directed set),

C

itself has a maximal element

d

by Zorn’s Lemma. We will show that

d

is the least

upper bound

of elments of

A

. It is clear that each

a

∈

A

is bounded above by

d

(

A

⊆

B

⊆

C

). If

t

is an upper bound of elements of

A

, then it is an upper bound of elements of

B

, and hence an upper bound of elements of

C

, which means

d

≤

t

.

(

2

.

⇒

1

.

)

By assumption

⋁

∅

exists (

=

0

), so that

⋀

L

=

0

. Now suppose

A

is a

proper subset

of

L

. We want to show that

⋀

A

exists. If

A

=

∅

, then

⋀

A

=

⋁

L

=

1

by definition of an arbitrary meet over the

empty set

. So assume

A

≠

∅

. Let

A

′

be the set of

lower bounds

of

A

:

A

′

=

{

x

∈

L

∣

x

≤

a

⁢

for all

⁢

a

∈

A

}

and let

b

=

⋁

A

′

, the least upper bound of

A

′

.

b

exists by assumption. Since

A

is a set of upper bounds of

A

′

,

b

≤

a

for all

a

∈

A

. This means that

b

is a lower bound of elements of

A

, or

b

∈

A

′

. If

x

is any lower bound of elements of

A

, then

x

≤

b

, since

x

is bounded above by

b

(

b

=

⋁

A

′

). This shows that

⋀

A

exists and is equal to

b

.
∎

Remarks

.

•

Dually, a poset is a complete lattice iff every subset has an

infimum

iff infimum exists for every finite subset and every directed subset.

•

The above proposition shows, for example, that every

closure system

is a complete lattice.

Title

criteria for a poset to be a complete lattice

Canonical

name

CriteriaForAPosetToBeACompleteLattice

Date of creation

2013-03-22 16:37:53

Last modified on

2013-03-22 16:37:53

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 06B23

Classification

msc 03G10

Related topic

MeetContinuous

Related topic

IntersectionStructure

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