complete semilattice

complete semilattice

A

complete

join-semilattice

is a join-semilattice

L

such that for any subset

A

⊆

L

,

⋁

A

, the arbitrary join

operation

on

A

, exists. Dually, a complete

meet-semilattice

is a meet-semilattice such that

⋀

A

exists for any

A

⊆

L

. Because there are no

restrictions

placed on the subset

A

, it turns out that a
complete join-semilattice is a complete meet-semilattice, and therefore a

complete lattice

. In other words, by dropping the arbitrary join (meet) operation from a complete lattice, we end up with nothing new. For a proof of this, see here (

http://planetmath.org/CriteriaForAPosetToBeACompleteLattice

). The crux of the matter lies in the fact that

⋁

(

⋀

) applies to

any

set, including

L

itself, and the

empty set

∅

, so that

L

always contains has a top and a bottom.

Variations

. To obtain new objects, one looks for variations in the definition of “complete”. For example, if we require that any

A

⊆

L

to be

countable

, we get what is a called a

countably complete join-semilattice

(or dually, a

countably complete meet-semilattice

). More generally, if

κ

is any

cardinal

, then a

κ

-complete join-semilattice is a

semilattice

L

such that for any set

A

⊆

L

such that

|

A

|

≤

κ

,

⋁

A

exists. If

κ

is finite, then

L

is just a join-semilattice. When

κ

=

∞

, the only requirement on

A

⊆

L

is that it be non-empty. In [1], a complete semilattice is defined to be a poset

L

such that for any non-empty

A

⊆

L

,

⋀

A

exists, and any

directed set

D

⊆

L

,

⋁

D

exists.

Example

. Let

A

and

B

be two

isomorphic

complete chains

(a chain that is a complete lattice) whose

cardinality

is

κ

. Combine the two chains to form a

lattice

L

by joining the top of

A

with the top of

B

, and the bottom of

A

with the bottom of

B

, so that

•

if

a

≤

b

in

A

, then

a

≤

b

in

L

•

if

c

≤

d

in

B

, then

c

≤

d

in

L

•

if

a

∈

A

,

c

∈

B

, then

a

≤

c

iff

a

is the bottom of

A

and

c

is the top of

B

•

if

a

∈

A

,

c

∈

B

, then

c

≤

a

iff

a

is the top of

A

and

c

is the bottom of

B

Now,

L

can be easily seen to be a

κ

-complete lattice. Next, remove the bottom element of

L

to obtain

L

′

. Since, the meet operation no longer works on all pairs of elements of

L

′

while

∨

still works,

L

′

is a join-semilattice that is not a lattice. In fact,

⋁

works on all subsets of

L

′

. Since

|

L

′

|

=

κ

, we see that

L

′

is a

κ

-complete join-semilattice.

Remark

. Although a complete semilattice is the same as a complete lattice, a

homomorphism

f

between, say, two complete join-semilattices

L

1

and

L

2

, may fail to be a homomorphism between

L

1

and

L

2

as complete lattices. Formally, a

complete join-semilattice homomorphism

between two complete join-semilattices

L

1

and

L

2

is a

function

f

:

L

1

→

L

2

such that for any subset

A

⊆

L

1

, we have

f

⁢

(

⋁

A

)

=

⋁

f

⁢

(

A

)

where

f

⁢

(

A

)

=

{

f

⁢

(

a

)

∣

a

∈

A

}

. Note that it is not required that

f

⁢

(

⋀

A

)

=

⋀

f

⁢

(

A

)

, so that

f

needs not be a

complete lattice homomorphism

.

To give a concrete example where a complete join-semilattice homomorphism

f

fails to be complete lattice homomorphism, take

L

from the example above, and define

f

:

L

→

L

by

f

⁢

(

a

)

=

1

if

a

≠

0

and

f

⁢

(

0

)

=

0

. Then for any

A

⊆

L

, it is evident that

f

⁢

(

⋁

A

)

=

⋁

f

⁢

(

A

)

. However, if we take two

incomparable

elements

a

,

b

∈

L

, then

f

⁢

(

a

∧

b

)

=

f

⁢

(

0

)

=

0

, while

f

⁢

(

a

)

∧

f

⁢

(

b

)

=

1

∧

1

=

1

.

References

1

G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove, D. S. Scott,

Continuous

Lattices and Domains

, Cambridge University Press, Cambridge (2003).

2

P. T. Johnstone,

Stone Spaces

, Cambridge University Press (1982).

Title

complete semilattice

Canonical name

CompleteSemilattice

Date of creation

2013-03-22 17:44:49

Last modified on

2013-03-22 17:44:49

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

13

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06A12

Classification

msc 06B23

Synonym

countably complete upper-semilattice

Synonym

countably complete lower-semilattice

Synonym

complete upper-semilattice homomorphism

Synonym

complete lower-semilattice homomorphism

Related topic

CompleteLattice

Related topic

Semilattice

Related topic

ArbitraryJoin

Defines

countably complete join-semilattice

Defines

countably complete meet-semilattice

Defines

complete join-semilattice homomorphism

Defines

complete meet-semilattice homomorphism

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