aarjaneiro/planetmath huggingface dataset:
normal subgroup lattice is modular

normal subgroup lattice is modular

The fact that the

normal subgroups

of a group

G

form a

lattice

(call it

N

⁢

(

G

)

) is proved here (

http://planetmath.org/NormalSubgroupsFormSublatticeOfASubgroupLattice

). The only remaining item is to show that

N

⁢

(

G

)

is

modular

(

http://planetmath.org/ModularLattice

). This means, for any normal subgroups

H

,

K

,

L

of

G

such that

L

⊆

K

,

L

∨

(

H

∧

K

)

=

(

L

∨

H

)

∧

K

,

where the meet

operation

A

∧

B

denotes set

intersection

A

∩

B

, and the join operation

A

∨

B

denotes the

subgroup

generated by

A

∪

B

.

Proof.

First, we show that

L

∨

(

H

∧

K

)

⊆

(

L

∨

H

)

∧

K

. It is easy to see that

1.

L

⊆

K

∧

(

L

∨

H

)

:

L

⊆

K

is assumed and

L

⊆

L

∨

H

follows from the definition of

∨

, and

2.

H

∧

K

⊆

K

∧

(

L

∨

H

)

:

H

∧

K

⊆

K

follows from the definition of

∧

, and

H

∧

K

⊆

H

⊆

L

∨

H

.

As a result,

L

∨

(

H

∧

K

)

⊆

K

∧

(

L

∨

H

)

=

(

L

∨

H

)

∧

K

.

Before proving the other

inclusion

, we shall derive a small lemma concerning

L

∨

H

where

L

,

H

are normal subgroups of

G

:

L

∨

H

=

{

ℓ

⁢

h

∣

ℓ

∈

L

⁢

and

⁢

h

∈

H

}

.

Proof.

One direction is obvious, so we will just show

L

∨

H

⊆

{

ℓ

⁢

h

∣

ℓ

∈

L

⁢

and

⁢

h

∈

H

}

. Any element of

L

∨

H

can be expressed as a finite

product

of

elements

from

L

or

H

. This finite product

representation

can be

reduced

so that no two

adjacent

elements belong to the same group. Next,

h

⁢

ℓ

=

(

h

⁢

ℓ

⁢

h

-

1

)

⁢

h

, where

h

⁢

ℓ

⁢

h

-

1

∈

L

since

L

is normal, showing that an

ℓ

on the right of the product can be “filtered” to the left of the product (of course, this “filtering” changes

ℓ

to another element of

L

, but it is the form of the product, not the elements in the product, that we are interested in). This implies that the finite product representation can further be reduced (by an

induction

argument

) so it has the final form

ℓ

⁢

h

.
∎

Now back to the main proof. Take any

g

∈

(

L

∨

H

)

∧

K

. Then

g

∈

K

and

g

∈

L

∨

H

and so

g

=

ℓ

⁢

h

for some

ℓ

∈

L

and

h

∈

H

by the lemma just shown. Since

g

∈

K

, this means

h

=

ℓ

-

1

⁢

g

∈

L

⁢

K

⊆

K

. So

h

∈

H

∧

K

. We have just expressed

g

as a product of

ℓ

∈

L

and

h

∈

H

∧

K

, and so

g

∈

L

∨

(

H

∧

K

)

.
∎

Title

normal subgroup lattice is modular

Canonical name

NormalSubgroupLatticeIsModular

Date of creation

2013-03-22 15:50:33

Last modified on

2013-03-22 15:50:33

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Derivation

Classification

msc 06C05

Classification

msc 20E25

Generated on Fri Feb 9 09:59:27 2018 by

LaTeXML
semimodular lattice

semimodular lattice

A

lattice

L

is

semimodular

1

1

Or

upper semimodular

, if one wants to stress the
distinction with lower semimodular lattices.

if for any

a

and

b

∈

L

,

a

∧

b

≺

a

implies

b

≺

a

∨

b

,

where

≺

denotes the covering relation in

L

.
Dually, a lattice

L

is said to be

lower semimodular

if for any

a

and

b

∈

L

,

b

≺

a

∨

b

implies

a

∧

b

≺

a

.

A

chain finite

lattice is

modular

(

http://planetmath.org/ModularLattice

)
if and only if it is both semimodular and lower semimodular.

The smallest lattice which is semimodular but not modular is

\xymatrix

⁢

&

⁢

1

⁢

\ar

⁢

@

-

[

l

⁢

d

]

⁢

\ar

⁢

@

-

[

d

]

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

&

⁢

a

⁢

\ar

⁢

@

-

[

d

]

⁢

&

⁢

b

⁢

\ar

⁢

@

-

[

l

⁢

d

]

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

&

⁢

c

⁢

\ar

⁢

@

-

[

d

]

⁢

d

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

&

⁢

&

⁢

e

⁢

\ar

⁢

@

-

[

l

⁢

d

]

⁢

&

⁢

0

⁢

&

since

d

≤

a

but

a

∧

(

c

∨

d

)

≠

(

a

∧

c

)

∨

d

.

Title

semimodular lattice

Canonical

name

SemimodularLattice

Date of creation

2013-03-22 15:26:20

Last modified on

2013-03-22 15:26:20

Owner

mps (409)

Last modified by

mps (409)

Numerical id

9

Author

mps (409)

Entry type

Definition

Classification

msc 06C10

Synonym

upper semimodular lattice

Synonym

lower semimodular lattice

Related topic

ModularLattice

Related topic

IncidenceGeometry

Generated on Fri Feb 9 09:59:29 2018 by

LaTeXML
commutativity relation in an orthocomplemented lattice

commutativity relation in an orthocomplemented lattice

Let

L

be an

orthocomplemented lattice

with

a

,

b

∈

L

. We say that

a

commutes

with

b

if

a

=

(

a

∧

b

)

∨

(

a

∧

b

⟂

)

. When

a

commutes with

b

, we write

a

⁢

C

⁡

b

. Dualize everything, we have that

a

dually commutes

with

b

, written

a

⁢

D

⁡

b

, if

a

=

(

a

∨

b

)

∧

(

a

∨

b

⟂

)

.

Some properties

. Below are some properties of the commutativity

relations

C

and

D

.

1.

C

is

reflexive

.

2.

a

⁢

C

⁡

b

iff

a

⁢

C

⁡

b

⟂

.

3.

a

⁢

C

⁡

b

iff

a

⟂

⁢

D

⁡

b

⟂

.

4.

if

a

≤

b

or

a

≤

b

⟂

, then

a

⁢

C

⁡

b

.

5.

a

is said to

orthogonally commute

with

b

if

a

⁢

C

⁡

b

and

b

⁢

C

⁡

a

. In this case, we write

a

⁢

M

⁡

b

. The terminology comes from the following fact:

a

⁢

M

⁡

b

iff there are

x

,

y

,

z

,

t

∈

L

, with:

(a)

x

⟂

y

(

x

is

orthogonal

to

y

, or

x

≤

y

⟂

),

(b)

z

⟂

t

,

(c)

x

⟂

z

,

(d)

a

=

x

∨

y

, and

(e)

b

=

z

∨

t

.

6.

C

is

symmetric

iff

D

=

C

(

=

M

)

iff

L

is an orthomodular lattice.

7.

C

is an

equivalence relation

iff

C

=

L

×

L

iff

L

is a

Boolean algebra

.

Remark

. More generally, one can define commutativity

C

on an orthomodular poset

P

: for

a

,

b

∈

P

,

a

⁢

C

⁡

b

iff

a

∧

b

,

a

∧

b

⟂

, and

(

a

∧

b

)

∨

(

a

∧

b

⟂

)

exist, and

(

a

∧

b

)

∨

(

a

∧

b

⟂

)

=

a

. Dual commutativity and mutual commutativity in an orthomodular poset are defined similarly (with the provision that the

binary operations

on the pair of

elements

are meaningful).

References

1

L. Beran,

Orthomodular Lattices, Algebraic Approach

, Mathematics and Its Applications (East European Series), D. Reidel Publishing Company, Dordrecht, Holland (1985).

Title

commutativity relation in an orthocomplemented lattice

Canonical name

CommutativityRelationInAnOrthocomplementedLattice

Date of creation

2013-03-22 16:43:22

Last modified on

2013-03-22 16:43:22

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06C15

Classification

msc 03G12

Defines

dually commute

Defines

orthogonally commute

Generated on Fri Feb 9 09:59:38 2018 by

LaTeXML
complemented lattice

complemented lattice

Let

L

be a

bounded lattice

(with

0

and

1

), and

a

∈

L

. A

complement

of

a

is an

element

b

∈

L

such that

a

∧

b

=

0

and

a

∨

b

=

1

.

Remark

. Complements may not exist. If

L

is a non-trivial chain, then no element (other than

0

and

1

) has a complement. This also shows that if

a

is a complement of a

non-trivial element

b

, then

a

and

b

form an

antichain

.

An element in a bounded lattice is

complemented

if it has a complement. A

complemented lattice

is a bounded lattice in which every element is complemented.

Remarks

.

•

In a complemented lattice, there may be more than one complement corresponding to each element. Two elements are said to be

related

, or

perspective

if they have a common complement. For example, the following

lattice

is complemented.

\xymatrix

⁢

&

⁢

1

⁢

\ar

⁢

@

-

[

l

⁢

d

]

⁢

\ar

⁢

@

-

[

d

]

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

&

⁢

a

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

&

⁢

b

⁢

\ar

⁢

@

-

[

d

]

⁢

&

⁢

c

⁢

\ar

⁢

@

-

[

l

⁢

d

]

⁢

&

⁢

0

⁢

&

Note that none of the non-trivial elements have unique complements. Any two non-trivial elements are related via the third.

•

If a complemented lattice

L

is a

distributive lattice

, then

L

is

uniquely complemented

(in fact, a

Boolean lattice

). For if

y

1

and

y

2

are two complements of

x

, then

y

2

=

1

∧

y

2

=

(

x

∨

y

1

)

∧

y

2

=

(

x

∧

y

2

)

∨

(

y

1

∧

y

2

)

=

0

∨

(

y

1

∧

y

2

)

=

y

1

∧

y

2

.

Similarly,

y

1

=

y

2

∧

y

1

. So

y

2

=

y

1

.

•

In the category of complemented lattices, a

morphism

between two objects is a

{

0

,

1

}

-

lattice homomorphism

; that is, a lattice homomorphism which preserves

0

and

1

.

Title

complemented lattice

Canonical name

ComplementedLattice

Date of creation

2013-03-22 15:02:25

Last modified on

2013-03-22 15:02:25

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

26

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06C15

Classification

msc 06B05

Synonym

perspective elements

Synonym

complemented

Related topic

Perspectivity

Related topic

OrthocomplementedLattice

Related topic

PseudocomplementedLattice

Related topic

DifferenceOfLatticeElements

Related topic

Pseudocomplement

Defines

related elements in lattice

Defines

complement

Defines

complemented element

Generated on Fri Feb 9 09:59:41 2018 by

LaTeXML
relative complement

relative complement

A

complement

of an

element

in a

lattice

is only defined when the lattice in question is

bounded

(

http://planetmath.org/BoundedLattice

). In general, a lattice is not bounded and there are no complements to speak of. Nevertheless, if the sublattice of a lattice is bounded, we can speak of complements of an element

relative

to that sublattice.

Let

L

be a lattice,

a

an element of

L

, and

I

=

[

b

,

c

]

an interval (

http://planetmath.org/LatticeInterval

) in

L

. An element

d

∈

L

is said to be a complement of

a

relative

to

I

if

a

∨

d

=

c

⁢

and

⁢

a

∧

d

=

b

.

It is easy to see that

a

≤

c

and

b

≤

a

,  so

a

∈

I

. Similarly,

d

∈

I

.

An element

a

∈

L

is said to be

relatively complemented

if for every interval

I

in

L

with

a

∈

I

, it has a complement relative to

I

. The lattice

L

itself is called a

relatively

complemented lattice

if every element of

L

is relatively complemented. Equivalently,

L

is relatively complemented iff each of its interval is a complemented lattice.

Remarks

.

•

A relatively complemented lattice is complemented if it is bounded.

Conversely

, a complemented lattice is relatively complemented if it is

modular

(

http://planetmath.org/ModularLattice

).

•

The notion of a

relative complement

of an element in a lattice has nothing to do with that found in

set theory

: let

U

be a set and

A

,

B

subsets of

U

, the relative complement of

A

in

B

is the set theoretic

difference

B

-

A

. While the relative difference is necessarily a subset of

B

,

A

does not have to be a subset of

B

.

Title

relative complement

Canonical

name

RelativeComplement

Date of creation

2013-03-22 15:51:45

Last modified on

2013-03-22 15:51:45

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

11

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06C15

Related topic

RelativePseudocomplement

Related topic

BrouwerianLattice

Defines

relatively complemented lattice

Defines

relatively complemented

Generated on Fri Feb 9 09:59:53 2018 by

LaTeXML
sectionally complemented lattice

sectionally complemented lattice

Proposition 1

.

Let

L

be a

lattice

with the least

element

0

. Then the following are

equivalent

:

1.

Every pair of elements have a

difference

(

http://planetmath.org/DifferenceOfLatticeElements

).

2.

for any

a

∈

L

, the

lattice interval

[

0

,

a

]

is a

complemented lattice

.

Proof.

Suppose first that every pair of elements have a difference. Let

b

∈

[

0

,

a

]

and let

c

be a difference between

a

and

b

. So

0

=

b

∧

c

and

c

∨

b

=

b

∨

a

=

a

, since

b

≤

a

. This shows that

c

is a

complement

of

b

in

[

0

,

a

]

.

Next suppose that

[

0

,

a

]

is complemented for every

a

∈

L

. Let

x

,

y

∈

L

be any two elements in

L

. Let

a

=

x

∨

y

. Since

[

0

,

a

]

is complemented,

y

has a complement, say

z

∈

[

0

,

a

]

. This means that

y

∧

z

=

0

and

y

∨

z

=

a

=

x

∨

y

. Therefore,

z

is a difference of

x

and

y

.
∎

Definition

. A lattice

L

with the least element

0

satisfying either of the two equivalent conditions above is called a

sectionally complemented lattice

.

Every relatively complemented lattice is sectionally complemented. Every sectionally complemented

distributive lattice

is relatively complemented.

Dually, one defines a

dually sectionally complemented lattice

to be a lattice

L

with the top element

1

such that for every

a

∈

L

, the

interval

[

a

,

1

]

is complemented, or, equivalently, the lattice dual

L

∂

is sectionally complemented.

References

1

G. Grätzer,

General

Lattice Theory

, 2nd Edition, Birkhäuser (1998)

Title

sectionally complemented lattice

Canonical name

SectionallyComplementedLattice

Date of creation

2013-03-22 17:58:46

Last modified on

2013-03-22 17:58:46

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06C15

Classification

msc 06B05

Related topic

DifferenceOfLatticeElements

Defines

sectionally complemented

Defines

dually sectionally complemented lattice

Generated on Fri Feb 9 10:00:04 2018 by

LaTeXML
Kurosh-Ore theorem

Kurosh-Ore theorem

Theorem 1

(Kurosh-Ore).

Let

L

be a

modular lattice

and suppose that

a

∈

L

has two

irredundant

decompositions of joins of

join-irreducible

elements

:

a

=

x

1

∨

⋯

∨

x

m

=

y

1

∨

⋯

∨

y

n

.

Then

1.

m

=

n

, and

2.

every

x

i

can be replaced by some

y

j

, so that

a

=

x

1

∨

⋯

∨

x

i

-

1

∨

y

j

∨

x

i

+

1

∨

⋯

∨

x

m

.

There is also a dual statement of the above

theorem

in terms of meets.

Remark

. Additionally, if

L

is a

distributive lattice

, then the second property above (known the

replacement property

) can be strengthened: each

x

i

is equal to some

y

j

. In other words, except for the re-ordering of elements in the decomposition, the above join is unique.

Title

Kurosh-Ore theorem

Canonical name

KuroshOreTheorem

Date of creation

2013-03-22 18:10:11

Last modified on

2013-03-22 18:10:11

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 06D05

Classification

msc 06C05

Classification

msc 06B05

Generated on Fri Feb 9 10:00:34 2018 by

LaTeXML
Kleene algebra

Kleene algebra

This entry concerns a

Kleene algebra

that is defined as a

lattice

satisfying certain conditions. There is another of Kleene algebra, which is the abstraction of the

algebra

of

regular expressions

in the theory of computations. The two

concepts

are different. For Kleene algebras of the second kind, please see this link (

http://planetmath.org/KleeneAlgebra

).

A lattice

L

is said to be a

Kleene algebra

if it is a

De Morgan algebra

(with the associated unary

operator

∼

on

L

) such that

(

∼

a

∧

a

)

≤

(

∼

b

∨

b

)

for all

a

,

b

∈

L

.

Any

Boolean algebra

A

is a Kleene algebra, if the

complementation

operator

′

is interpreted as

∼

. This is true because

a

′

∧

a

=

0

≤

1

=

b

′

∨

b

for all

a

,

b

∈

A

. The

converse

is not true. For example, consider the chain

𝐧

=

{

0

,

1

,

…

,

n

}

, with the usual

ordering

. Define

∼

by

∼

(

k

)

=

n

-

k

. Then it is easy to see that

∼

satisfies

all the defining conditions of a De Morgan algebra. In

addition

, since every

a

,

b

∈

𝐧

are

comparable

, say

a

≤

b

, then

(

∼

a

∧

a

)

≤

a

≤

b

≤

(

∼

b

∨

b

)

. And if

b

≤

a

on the other hand, then

∼

a

≤

∼

b

so that

(

∼

a

∧

a

)

≤

∼

a

≤

∼

b

≤

(

∼

b

∨

b

)

. But

𝐧

is not

Boolean

, as

a

∨

b

is never

n

unless one of them is.

Remark

. As Boolean algebras are the algebraic realizations of the classical two-valued

propositional logic

, Kleene algebras are the realizations of a three-valued propositional logic, where the three truth values can be described as true (

2

), false (

0

), and unknown (

1

). Just as

{

0

,

1

}

is the simplest Boolean algebra (it is a simple algebra),

{

0

,

1

,

2

}

is the simplest Kleene algebra, where

∼

is defined the same way as in the example above.

References

1

G. Grätzer,

General

Lattice Theory

, 2nd Edition, Birkhäuser (1998)

Title

Kleene algebra

Canonical name

KleeneAlgebra1

Date of creation

2013-03-22 17:08:43

Last modified on

2013-03-22 17:08:43

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06D30

Related topic

KleeneAlgebra

Generated on Fri Feb 9 10:01:38 2018 by

LaTeXML
Ockham algebra

Ockham algebra

A

lattice

L

is called an

Ockham algebra

if

1.

L

is

distributive

2.

L

is

bounded

, with

0

as the bottom and

1

as the top

3.

there is a unary

operator

¬

on

L

with the following properties:

(a)

¬

satisfies

the de Morgan’s laws; this means that:

*

¬

⁡

(

a

∨

b

)

=

¬

⁢

a

∧

¬

⁢

b

and

*

¬

⁡

(

a

∧

b

)

=

¬

⁢

a

∨

¬

⁢

b

(b)

¬

⁢

0

=

1

and

¬

⁢

1

=

0

Such a unary operator is an example of a dual

endomorphism

. When applied,

¬

interchanges the

operations

of

∨

and

∧

, and

0

and

1

.

An Ockham algebra is a

generalization

of a

Boolean algebra

, in the sense that

¬

replaces

′

, the

complement

operator, on a Boolean algebra.

Remarks

.

•

An intermediate

concept

is that of a

De Morgan algebra

, which is an Ockham algebra with the additional requirement that

¬

⁡

(

¬

⁢

a

)

=

a

.

•

In the

category

of Ockham algebras, the

morphism

between any two objects is a

{

0

,

1

}

-

lattice homomorphism

(

http://planetmath.org/LatticeHomomorphism

)

f

that preserves

¬

:

f

⁢

(

¬

⁢

a

)

=

¬

⁢

f

⁢

(

a

)

. In fact,

f

⁢

(

0

)

=

f

⁢

(

¬

⁢

1

)

=

¬

⁢

f

⁢

(

1

)

=

¬

⁢

1

=

0

, so that it is safe to drop the

assumption

that

f

preserves

0

.

References

1

T.S. Blyth, J.C. Varlet,

Ockham Algebras

, Oxford University Press, (1994).

2

T.S. Blyth,

Lattices and Ordered

Algebraic Structures

, Springer, New York (2005).

Title

Ockham algebra

Canonical name

OckhamAlgebra

Date of creation

2013-03-22 17:08:34

Last modified on

2013-03-22 17:08:34

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06D30

Generated on Fri Feb 9 10:01:47 2018 by

LaTeXML
distributive inequalities

distributive inequalities

Let

L

be a

lattice

. Then for

a

,

b

,

c

∈

L

, we have the following

inequalities

:

1.

a

∨

(

b

∧

c

)

≤

(

a

∨

b

)

∧

(

a

∨

c

)

,

2.

(

a

∧

b

)

∨

(

a

∧

c

)

≤

a

∧

(

b

∨

c

)

.

Proof.

Since

a

≤

a

∨

b

and

a

≤

a

∨

c

,

a

≤

(

a

∨

b

)

∧

(

a

∨

c

)

.
Similarly,

b

∧

c

≤

b

≤

a

∨

b

and

b

∧

c

≤

c

≤

a

∨

c

imply

b

∧

c

≤

(

a

∨

b

)

∧

(

a

∨

c

)

.
Together, we have

a

∨

(

b

∧

c

)

≤

(

a

∨

b

)

∧

(

a

∨

c

)

.

The second inequality is the dual of the first one.
∎

The two inequalities above are called the

distributive inequalities

.

Proposition

A lattice

L

is a

distributive lattice

if one of the following inequalities holds:

1.

(

a

∨

b

)

∧

(

a

∨

c

)

≤

a

∨

(

b

∧

c

)

,

2.

a

∧

(

b

∨

c

)

≤

(

a

∧

b

)

∨

(

a

∧

c

)

.

Proof.

By the distributive inequalities, all we need to show is that 1. implies 2. (that 2. implies 1. is just the dual statement). So suppose 1. holds. Then

(

a

∧

b

)

∨

(

a

∧

c

)

≥

(

(

a

∧

b

)

∨

a

)

∧

(

(

a

∧

b

)

∨

c

)

⁢

by assumption

=

a

∧

(

(

a

∧

b

)

∨

c

)

⁢

by absorption

≥

a

∧

(

(

c

∨

a

)

∧

(

c

∨

b

)

)

⁢

by assumption

=

(

a

∧

(

c

∨

a

)

)

∧

(

c

∨

b

)

⁢

meet associativity

=

a

∧

(

c

∨

b

)

.

⁢

by absorption

∎

Title

distributive inequalities

Canonical name

DistributiveInequalities

Date of creation

2013-03-22 16:37:48

Last modified on

2013-03-22 16:37:48

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Derivation

Classification

msc 06D99

Related topic

ModularInequality

Generated on Fri Feb 9 10:01:57 2018 by

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distributive lattice

distributive lattice

A

lattice

(

http://planetmath.org/Lattice

) is said to be

distributive

if it satisifes either (and therefore both) of the distributive laws (

http://planetmath.org/Distributive

):

•

x

∧

(

y

∨

z

)

=

(

x

∧

y

)

∨

(

x

∧

z

)

•

x

∨

(

y

∧

z

)

=

(

x

∨

y

)

∧

(

x

∨

z

)

Every

distributive lattice

is

modular

(

http://planetmath.org/ModularLattice

).

Examples of distributive lattices include

Boolean

lattices (

http://planetmath.org/BooleanLattice

),

totally ordered sets

, and the

subgroup

lattices (

http://planetmath.org/LatticeOfSubgroups

) of

locally cyclic groups

.

Title

distributive lattice

Canonical

name

DistributiveLattice

Date of creation

2013-03-22 12:27:23

Last modified on

2013-03-22 12:27:23

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

20

Author

yark (2760)

Entry type

Definition

Classification

msc 06D99

Related topic

Distributive

Related topic

Lattice

Related topic

BirkhoffPrimeIdealTheorem

Generated on Fri Feb 9 10:01:59 2018 by

LaTeXML
distributivity

distributivity

Given a set (

http://planetmath.org/Set

)

S

with two

binary operations

+

:

S

×

S

→

S

and

⋅

:

S

×

S

→

S

, we say that

⋅

is

right distributive

over

+

if

(

a

+

b

)

⋅

c

=

(

a

⋅

c

)

+

(

b

⋅

c

)

⁢

for

⁢

all

⁢

a

,

b

,

c

∈

S

and

left distributive

over

+

if

a

⋅

(

b

+

c

)

=

(

a

⋅

b

)

+

(

a

⋅

c

)

⁢

for

⁢

all

⁢

a

,

b

,

c

∈

S

.

If

⋅

is both left and right distributive over

+

, then it is said to be

distributive

over

+

(or, alternatively, we may say that

⋅

distributes over

+

).

Title

distributivity

Canonical

name

Distributivity

Date of creation

2013-03-22 13:47:00

Last modified on

2013-03-22 13:47:00

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

15

Author

yark (2760)

Entry type

Definition

Classification

msc 06D99

Classification

msc 16-00

Classification

msc 13-00

Classification

msc 17-00

Synonym

distributive law

Synonym

distributive property

Related topic

Ring

Related topic

DistributiveLattice

Related topic

NearRing

Defines

distributive

Defines

left distributive

Defines

right distributive

Defines

left-distributive

Defines

right-distributive

Defines

distributes over

Defines

left distributivity

Defines

right distributivity

Defines

left distributes over

Defines

left distributive law

Defines

right distributive law

Generated on Fri Feb 9 10:02:00 2018 by

LaTeXML