lattice homomorphism

lattice homomorphism

Let

L

and

M

be lattices. A map

ϕ

from

L

to

M

is called a

lattice homomorphism

if

ϕ

respects meet and join. That is, for

a

,

b

∈

L

,

•

ϕ

⁢

(

a

∧

b

)

=

ϕ

⁢

(

a

)

∧

ϕ

⁢

(

b

)

, and

•

ϕ

⁢

(

a

∨

b

)

=

ϕ

⁢

(

a

)

∨

ϕ

⁢

(

b

)

.

From this definition, one also defines

lattice isomorphism

,

lattice endomorphism

,

lattice automorphism

respectively, as a

bijective

lattice homomorphism, a lattice homomorphism into itself, and a lattice isomorphism onto itself.

If in

addition

L

is a

bounded lattice

with top

1

and bottom

0

, with

ϕ

and

M

defined as above, then

ϕ

⁢

(

a

)

=

ϕ

⁢

(

1

∧

a

)

=

ϕ

⁢

(

1

)

∧

ϕ

⁢

(

a

)

, and

ϕ

⁢

(

a

)

=

ϕ

⁢

(

0

∨

a

)

=

ϕ

⁢

(

0

)

∨

ϕ

⁢

(

a

)

for all

a

∈

L

. Thus

L

is mapped onto a

bounded

sublattice

ϕ

⁢

(

L

)

of

M

, with top

ϕ

⁢

(

1

)

and bottom

ϕ

⁢

(

0

)

.

If both

L

and

M

are bounded with lattice homomorphism

ϕ

:

L

→

M

, then

ϕ

is said to be a

{

0

,

1

}

-

lattice homomorphism

if

ϕ

⁢

(

1

)

and

ϕ

⁢

(

0

)

are top and bottom of

M

. In other words,

ϕ

⁢

(

1

L

)

=

1

M

and

ϕ

⁢

(

0

L

)

=

0

M

,

where

1

L

,

1

M

,

0

L

,

0

M

are top and bottom elements of

L

and

M

respectively.

Remarks.

•

The idea behind these definitions comes from the idea of a

homomorphism

between two

algebraic systems

of the same type. We require the the homomorphism to preserve all finitary operations, including the nullary ones. This means that if the algebraic system contains

constants

, they need to be preserved under the homomorphism. Thus, if

L

and

M

are both bounded lattices, a homomorphism between

L

and

M

must preserve

0

and

1

. Similarly, if

L

only has

0

and

M

is bounded, then a homomorphism between them should preserve

0

alone.

•

In the case of

complete

lattices, there are

operations

that are infinitary, so the homomorphism between two complete lattices should preserve the infinitary operations as well. The resulting lattice homomorphism is a

complete lattice homomorphism

.

•

One can show that every

Boolean algebra

B

can be embedded into the

power set

of some set

S

. That is, there is a

one-to-one

lattice homomorphism

ϕ

from

B

into a

Boolean subalgebra

of

2

S

(under the usual set union and set

intersection

operations) (see link below). If

B

is in addition a

complete lattice

and an

atomic lattice

, then

B

is

lattice

isomorphic

to

2

S

for some set

S

.

Title

lattice homomorphism

Canonical name

LatticeHomomorphism

Date of creation

2013-03-22 15:41:31

Last modified on

2013-03-22 15:41:31

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

13

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06B05

Classification

msc 06B99

Related topic

OrderPreservingMap

Related topic

RepresentingABooleanLatticeByFieldOfSets

Defines

lattice isomorphism

Defines

lattice endomorphism

Defines

lattice automorphism

Defines

Defines

1}-latticehomomorphism

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