arbitrary join

arbitrary join

Let

P

be a poset and

A

⊆

P

. The

join

of

A

is the

supremum

of

A

, if it exists. It is denoted by

⋁

A

or

⋁

i

∈

I

a

i

or

⋁

{

a

i

∣

i

∈

I

}

,

if the elements of

A

are

indexed by

a set

I

:

A

=

{

a

i

∣

i

∈

I

}

.

In other words,

⋁

A

=

sup

⁡

A

, where the

equality

is directed in the sense that one side is defined iff the other side is, and when this is the case, both sides are equal. Dually, one defines the

meet

of

A

to be the

infimum

of

A

, if it exists. The meet of

A

is denoted by

⋀

A

.

Remark

. The

concepts

of

⋁

and

sup

of an

ordered set

are identical. Besides being notationally distinct,

⋁

is often used in order theory, while

sup

is more prevalent in

analysis

. Moreover,

⋁

is generally being viewed as a (partial)

function

on the

powerset

2

P

of the poset

P

, while

sup

is frequently seen as an

operation

on

sequences

(or more generally nets) of elements of

P

.

1.

As was remarked, given a poset

P

, let us view

⋁

:

2

P

→

P

as a

partial function

. Then

⋁

is defined for all

singletons

. In fact

⋁

{

a

}

=

a

for all

a

∈

P

. Dually,

⋀

{

a

}

=

a

.

2.

If

⋁

is defined for all doubletons, then it is defined for all finite subsets of

P

. In this case,

P

is called a

join-semilattice

. Dually,

P

is a

meet-semilattice

is

⋀

is defined for all doubletons.

3.

If

A

⊆

B

, then

⋁

A

≤

⋁

B

, provided that both joins exist. We also have a dual statement:

A

⊆

B

implies that

⋀

B

≤

⋀

A

, provided that both meets exist.

4.

⋁

∅

exists iff

P

has a bottom

0

, and when this is the case,

⋁

∅

=

0

. This is essentially the result of the previous bulleted statement. Dually,

P

has a top

1

iff

⋀

∅

exists, and when this is the case

⋀

∅

=

1

.

5.

Simiarly

⋁

P

=

1

and

⋀

P

=

0

, where the equality is directed on both sides.

6.

It can be shown that if

⋁

is a

total function

from

2

P

to

P

, then

P

is a

complete lattice

. (see proof here (

http://planetmath.org/CriteriaForAPosetToBeACompleteLattice

)).

7.

Let

P

be a poset such that

⋁

is defined for all subsets (of

P

) of

cardinality

𝔪

, is it true that

⋁

is defined for all subsets of cardinality

≤

𝔪

? The answer is no, even when

𝔪

is finite. A

counterexample

can be constructed as follows.

\xymatrix

⁢

@

!

=

1

⁢

p

⁢

t

⁢

&

⁢

1

⁢

\ar

⁢

@

.

[

d

]

⁢

\ar

⁢

@

-

[

r

⁢

r

⁢

d

⁢

d

]

⁢

&

⁢

&

⁢

&

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

\ar

⁢

@

-

[

l

⁢

d

]

⁢

&

⁢

&

⁢

a

⁢

&

⁢

&

⁢

b

⁢

&

⁢

c

Let

C

be an

infinite chain

with a top

element

1

(this can be found by taking the set of natural numbers and dualize the usual order). Adjoin three elements

a

,

b

,

c

to

C

so that

a

and

b

are below all elements of

C

, and

c

is covered by

1

, and no two of

a

,

b

,

c

are

comparable

. This new poset

P

has the

property

that any three distinct elements have a join. For example,

⋁

{

a

,

b

,

c

}

=

1

. However,

⋁

{

a

,

b

}

does not exist.

Title

arbitrary join

Canonical

name

ArbitraryJoin

Date of creation

2013-03-22 17:27:53

Last modified on

2013-03-22 17:27:53

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06A06

Related topic

CompleteLattice

Related topic

CompleteSemilattice

Defines

arbitrary meet

Defines

infinite join

Defines

infinite meet

Generated on Fri Feb 9 09:50:44 2018 by

LaTeXML