properties of arbitrary joins and meets

properties of arbitrary joins and meets

In this entry, we list and prove some of the basic

properties of arbitrary joins and meets

. Some of the properties work in general posets, while others work only in lattices, and sometimes only in

Boolean algebras

.

Let

P

be a poset and

B

and

C

are subsets of

P

such that

⋁

B

and

⋁

C

exist.

1.

b

≤

⋁

B

for any

b

∈

B

. More generally, if

A

⊆

B

and

⋁

A

exists, then

⋁

A

≤

⋁

B

.

2.

if

b

≤

a

for every

b

∈

B

, then

⋁

B

≤

a

.

3.

If

B

=

⋃

{

B

i

∣

i

∈

I

}

, and each

b

i

:=

⋁

B

i

exists, then

⋁

{

b

i

∣

i

∈

I

}

exists and is equal to

⋁

B

.

Conversely

, if we drop the

assumption

that

⋁

B

exists, but assume instead that

⋁

{

b

i

∣

i

∈

I

}

exists, then

⋁

B

exists and is equal to

⋁

{

b

i

∣

i

∈

I

}

.

Proof.

Let

b

=

⋁

B

. For each

i

∈

I

, and each

c

∈

B

i

⊆

B

, we clearly have that

c

≤

b

. So

b

i

≤

b

, or that

b

is an

upper bound

of the

collection

D

:=

{

b

i

∣

i

∈

I

}

. If

d

is any upper bound of

D

, then

b

i

≤

d

. For any

c

∈

B

,

c

∈

B

i

for some

i

∈

I

, so that

c

≤

b

i

and hence

c

≤

d

. This shows that

b

≤

d

, or that

b

is the least upper bound of

D

.

Conversely, suppose

D

:=

⋁

{

b

i

∣

i

∈

I

}

exists and is equal to

d

. Then for any

b

∈

B

,

b

∈

B

i

for some

i

∈

I

, so that

b

≤

b

i

, and hence

b

≤

d

. This shows that

d

is an upper bound of

B

. If

f

is any upper bound of

B

, then

f

is an upper bound of

B

i

in particular, so

b

i

≤

f

. Since

i

is arbitray,

d

≤

f

, or that

d

is the least upper bound of

B

.
∎

4.

If

⋁

{

a

∨

b

∣

b

∈

B

}

exists, then it is equal to

a

∨

⋁

B

.

Proof.

Let

c

=

⋁

B

and

d

=

⋁

{

a

∨

b

∣

b

∈

B

}

. We want to show that

a

∨

c

=

d

. Since

b

≤

c

for all

b

∈

B

, we have that

a

∨

b

≤

a

∨

c

, and so

d

≤

a

∨

c

as

d

is the least upper bound of

{

a

∨

b

∣

b

∈

B

}

. On the other hand

a

∨

b

≤

d

, so that

a

≤

d

and

b

≤

d

, for all

b

∈

B

, the last

inequality

means that

c

≤

d

as well. Therefore

a

∨

c

≤

d

, and we are done.
∎

5.

If

P

is a Boolean algebra then the following hold:

(a)

⋀

(

B

′

)

exists, where

B

′

:=

{

b

′

∣

b

∈

B

}

, and is equal to

(

⋁

B

)

′

.

Proof.

Let

c

=

⋁

B

. Then

b

≤

c

for any

b

∈

B

, so that

c

′

≤

b

′

, or

c

′

is a

lower bound

for

B

′

. If

d

is any lower bound of

B

′

, then

d

≤

b

′

for every

b

∈

B

, so that

b

≤

d

′

, which implies

c

≤

d

′

, or

d

≤

c

′

. This means that

c

′

is the

greatest lower bound

of

B

′

, or that

c

′

=

⋀

(

B

′

)

.
∎

(b)

⋁

{

a

∧

b

∣

b

∈

B

}

exists and is equal to

a

∧

⋁

B

for any

a

∈

A

.

Proof.

Let

c

=

⋁

B

. Then

b

≤

c

for any

b

∈

B

and so

a

∧

b

≤

a

∧

c

. Therefore

a

∧

c

is an upper bound of

{

a

∧

b

∣

b

∈

B

}

. Now, if

d

is an upper bound of

{

a

∧

b

∣

b

∈

B

}

, then

a

∧

b

≤

d

for every

b

∈

B

. So

b

=

(

a

′

∨

a

)

∧

b

=

(

a

′

∧

b

)

∨

(

a

∧

b

)

≤

(

a

′

∧

b

)

∨

d

≤

a

′

∨

d

. This means that

a

′

∧

d

is an upper bound of

B

, so

c

≤

a

′

∨

d

. Therefore,

a

∧

c

≤

a

∧

(

a

′

∨

d

)

=

(

a

∧

a

′

)

∨

(

a

∧

d

)

=

a

∧

d

. Hence,

a

∧

c

is the least upper bound of

{

a

∧

b

∣

b

∈

B

}

.
∎

(c)

Define

B

∧

C

:=

{

b

∧

c

∣

b

∈

B

⁢

and

⁢

c

∈

C

}

.

Then

⋁

(

B

∧

C

)

exists and is equal to

⋁

B

∧

⋁

C

.

Proof.

Let

d

=

⋁

B

and

e

=

⋁

C

. Then

⋁

B

∧

⋁

C

=

d

∧

⋁

C

=

⋁

{

d

∧

c

∣

c

∈

C

}

by 4.b above. Now,

d

∧

c

=

⋁

B

∧

c

=

⋁

{

b

∧

c

∣

b

∈

B

}

again by 4.b. For each

c

∈

C

, set

B

c

:=

{

b

∧

c

∣

b

∈

B

}

. Then

⋁

B

c

=

d

∧

c

and

B

∧

C

=

⋃

{

B

c

∣

c

∈

C

}

. Therefore, by (3),

⋁

(

B

∧

C

)

exists and is equal to

⋁

{

⋁

B

c

∣

c

∈

C

}

=

⋁

{

d

∧

c

∣

c

∈

C

}

=

⋁

B

∧

⋁

C

.
∎

Remarks

.

•

All of the properties above can be dualized: assume that

B

and

C

are subsets of a poset

P

such that

⋀

B

and

⋀

C

exist, then:

(a)

if

A

⊆

B

and

⋀

A

exists, then

⋀

B

≤

⋀

A

.

(b)

if

a

≤

b

for every

b

∈

B

, then

a

≤

⋀

B

.

(c)

if

B

=

⋃

{

B

i

∣

i

∈

I

}

, and each

b

i

:=

⋀

B

i

exists, then

⋀

{

b

i

∣

i

∈

I

}

exists iff

⋀

B

does, and they are equal when one exists.

(d)

if

⋀

{

a

∧

b

∣

b

∈

B

}

exists, then it is equal to

a

∧

⋀

B

.

(e)

If

P

is a Boolean algebra, then

i.

⋁

(

B

′

)

exists, where

B

′

:=

{

b

′

∣

b

∈

B

}

, and is equal to

(

⋀

B

)

′

.

ii.

⋀

{

a

∨

b

∣

b

∈

B

}

exists and is equal to

a

∨

⋀

B

for any

a

∈

A

.

iii.

Define

B

∨

C

:=

{

b

∨

c

∣

b

∈

B

⁢

and

⁢

c

∈

C

}

.

Then

⋀

(

B

∨

C

)

exists and is equal to

⋀

B

∨

⋀

C

.

•

Notice that for property 5 above, the condition that

P

be

Boolean

can not be dropped. For example, consider the set

P

of non-negative integers. For any two

elements

a

,

b

∈

P

, define

a

≤

b

by the divisibility

relation

a

|

b

. It is easy to see that

P

is a bounded

distributive lattice

, with top element

0

and bottom element

1

. However, it is not

complemented

(suppose

2

′

is a

complement

of

2

, then

2

′

∧

2

=

1

, so that

2

′

must be odd, but then

2

′

∨

2

=

2

⋅

2

′

≠

0

, a

contradiction

).

More generally, for any subset

A

of

P

, define

⋁

A

to be the smallest non-negative integer

c

such that

a

|

c

for all

a

∈

A

, while

⋀

A

is the largest non-negative integer

d

such that

d

|

a

for all

a

∈

A

. If

A

=

∅

, define

⋁

A

=

1

and

⋀

A

=

0

. Then it is not hard to see that

P

is in

addition

a

complete lattice

. However, if we take

A

to be the set of all odd prime numbers, then

⋁

A

=

0

, so that for any

x

∈

P

,

x

∧

⋁

A

=

0

. But if

x

is any element in

A

, then

⋁

{

x

∧

a

∣

a

∈

A

}

=

x

≠

0

.

Title

properties of arbitrary joins and meets

Canonical name

PropertiesOfArbitraryJoinsAndMeets

Date of creation

2013-03-22 17:52:38

Last modified on

2013-03-22 17:52:38

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

12

Author

CWoo (3771)

Entry type

Derivation

Classification

msc 06A06

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