algebraic lattice

algebraic lattice

A

lattice

L

is said to be an

algebraic lattice

if it is a

complete lattice

and every element of

L

can be written as a join of

compact elements

.

As the name (G. Birkhoff originally coined the term) suggests, algebraic lattices are mostly found in lattices of

subalgebras of algebraic systems

. Below are some common examples.

Examples

.

1.

Groups. The lattice

L

⁢

(

G

)

of subgroups of a group

G

is known to be

complete

. Cyclic subgroups are compact elements of

L

⁢

(

G

)

. Since every subgroup

H

of

G

is the join of cyclic subgroups, each generated by an

element

h

∈

H

,

L

⁢

(

G

)

is algebraic.

2.

Vector spaces. The lattice

L

⁢

(

V

)

of

subspaces

of a vector space

V

is complete. Since each subspace has a basis, and since each element generates a one-dimensional subspace which is clearly

compact

,

L

⁢

(

V

)

is algebraic.

3.

Rings. The lattice

L

⁢

(

R

)

of ideals of a ring

R

is also complete, the join of a set of ideals of

R

is the

ideal generated by

elements in each of the ideals in the set. Any ideal

I

is the join of cyclic ideals generated by elements

r

∈

I

. So

L

⁢

(

R

)

is algebraic.

4.

Modules. The above two examples can be combined and generalized into one, the lattice

L

⁢

(

M

)

of submodules of a module

M

over a ring. The

arguments

are

similar

.

5.

Topological spaces

. The lattice of

closed subsets

of a topological space is in general

not

algebraic. The simplest example is

ℝ

with the

open intervals

forming the

subbasis

. To begin with, it is not complete: the union of closed subsets

[

0

,

1

-

1

n

]

,

n

∈

ℕ

is

[

0

,

1

)

, not a

closed set

. In

addition

,

ℝ

itself is a closed subset that is not compact.

Remarks

.

•

Since every element in an algebraic lattice is a join of compact elements, it is easy to see that every atom is compact: for if

a

is an atom in an algebraic lattice

L

, and

a

=

⋁

S

, where

S

⊆

L

is a set of compact elements

s

∈

L

, then each

s

is either

0

or

a

. Therefore,

S

consists of at most two elements

0

and

a

. But

S

can’t be a

singleton

consisting of

0

(otherwise

⋁

S

=

0

≠

a

), so

a

∈

S

and therefore

a

is compact.

•

The notion of being algebraic in a lattice can be generalized to an arbitrary dcpo: an

algebraic dcpo

is a dcpo

D

such that every

a

∈

D

can be written as

a

=

⋁

C

, where

C

is a

directed set

(in

D

) such that each element in

C

is compact.

References

1

G. Birkhoff

Lattice Theory

, 3rd Edition, AMS Volume XXV, (1967).

2

G. Grätzer,

General Lattice Theory

, 2nd Edition, Birkhäuser (1998).

3

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).

4

S. Vickers,

Topology

via Logic

, Cambridge University Press, Cambridge (1989).

Title

algebraic lattice

Canonical

name

AlgebraicLattice

Date of creation

2013-03-22 15:56:31

Last modified on

2013-03-22 15:56:31

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

15

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06B23

Classification

msc 51D25

Synonym

compactly-generated lattice

Related topic

SumOfIdeals

Defines

algebraic dcpo

Generated on Fri Feb 9 09:56:24 2018 by

LaTeXML