aarjaneiro/planetmath huggingface dataset:
join-infinite distributive

join-infinite distributive

A

lattice

L

is said to be

join-infinite distributive

if it is

complete

, and for any

element

x

∈

L

and any subset

M

of

L

, we have

x

∧

⋁

M

=

⋁

{

x

∧

y

∣

y

∈

M

}

.

(1)

Equation (1) is called the

join-infinite identity

, or

JID

for short. We also call

L

a JID lattice.

If

M

is any two-element set, then we see that the equation above is just one of the

distributive laws

, and hence any JID lattice is distributive. The

converse

of this statement is false. For example, take the set

N

of non-negative integers ordered by division, that is,

a

≤

b

iff

a

∣

b

. Then

N

is a

distributive lattice

. However,

N

fails JID, for if

M

is the set of all odd primes, then

⋁

M

=

0

, so

2

∧

(

⋁

M

)

=

2

, where as

⋁

{

2

∧

p

∣

p

∈

M

}

=

⋁

{

1

}

=

1

≠

2

.

Also any completely distributive lattice is JID. The converse of this is also false. For an example of a JID lattice that is not

completely distributive

, see the last paragraph below before the remarks.

Dually, a lattice

L

is said to be

meet-infinite distributive

if it is complete, and for any element

x

∈

L

and any subset

M

of

L

, we have

x

∨

⋀

M

=

⋀

{

x

∨

y

∣

y

∈

M

}

.

(2)

Equation (2) is called the

meet-infinite identity

, or

MID

for short.

L

is also called a MID lattice.

Now, unlike the case with a distributive lattice, where one distributive law implies its dual, JID does not necessarily imply MID, and vice versa. An example of a lattice satisfying MID but not JID can be found here (

http://planetmath.org/CompleteDistributivity

). The dual of this lattice then

satisfies

JID but not MID, and therefore is an example of a JID lattice that is not completely distributive. When a lattice is both join-infinite and meet-infinite distributive, it is said to be

infinite distributive

.

Remarks

•

It can be shown that any complete

Boolean lattice

is infinite distributive.

•

An intermediate

concept

between distributivity and infinite-distributivity is that of countable-distributivity: a lattice is

join-countable distributive

if JID holds for all

countable

subsets

M

of

L

, and

meet-countable distributive

if MID holds for all countable

M

⊆

L

.

•

When the sets

M

in JID are restricted to

filtered sets

, then the lattice

L

is

join continuous

. When

M

are

directed sets

in MID, then

L

is meet continuous.

Title

join-infinite distributive

Canonical name

JoininfiniteDistributive

Date of creation

2013-03-22 19:13:48

Last modified on

2013-03-22 19:13:48

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06D99

Synonym

JID

Synonym

MID

Related topic

MeetContinuous

Related topic

CompleteDistributivity

Defines

meet-infinite distributive

Defines

join-infinite identity

Defines

meet-infinite identity

Defines

infinite distributive

Defines

countably distributive

Defines

join-countable distributive

Defines

meet-countable distributive

Generated on Fri Feb 9 10:02:09 2018 by

LaTeXML
Boolean lattice

Boolean lattice

In this entry, the notions of a

Boolean lattice

, a

Boolean algebra

, and a

Boolean ring

are defined, compared and contrasted.

Boolean Lattices

A

Boolean lattice

B

is a

distributive lattice

in which for each

element

x

∈

B

there exists a

complement

x

′

∈

B

such that

x

∧

x

′

=

0

x

∨

x

′

=

1

(

x

′

)

′

=

x

(

x

∧

y

)

′

=

x

′

∨

y

′

(

x

∨

y

)

′

=

x

′

∧

y

′

In other words, a Boolean lattice is the same as a

complemented

distributive lattice. A

morphism

between two

Boolean

lattices is just a

lattice homomorphism

(so that

0

,

1

and

′

may not be preserved).

Boolean Algebras

A Boolean algebra is a Boolean lattice such that

′

and

0

are considered as

operators

(unary and nullary respectively) on the

algebraic system

. In other words, a morphism (or a

Boolean algebra homomorphism

) between two Boolean algebras must preserve

0

,

1

and

′

. As a result, the

category

of Boolean algebras and the category of Boolean lattices are not the same (and the former is a

subcategory

of the latter).

Boolean Rings

A

Boolean ring

is an (

associative

)

unital ring

R

such that for any

r

∈

R

,

r

2

=

r

. It is easy to see that

•

any Boolean ring has

characteristic

2

, for

2

⁢

r

=

(

2

⁢

r

)

2

=

4

⁢

r

2

=

4

⁢

r

,

•

and hence a commutative ring, for

a

+

b

=

(

a

+

b

)

2

=

a

2

+

a

⁢

b

+

b

⁢

a

+

b

2

=

a

+

a

⁢

b

+

b

⁢

a

+

b

, so

0

=

a

⁢

b

+

b

⁢

a

, and therefore

a

⁢

b

=

a

⁢

b

+

a

⁢

b

+

b

⁢

a

=

b

⁢

a

.

Boolean rings (with

identity

, but allowing 0=1) are

equivalent

to Boolean lattices. To view a Boolean ring as a Boolean lattice, define

x

∧

y

=

x

⁢

y

,

x

∨

y

=

x

+

y

+

x

⁢

y

,

and

x

′

=

1

+

x

.

To view a Boolean lattice as a Boolean ring, define

x

⁢

y

=

x

∧

y

and

x

+

y

=

(

x

′

∧

y

)

∨

(

x

∧

y

′

)

.

The category of Boolean algebras is

naturally equivalent

to the category of Boolean rings.

References

1

G. Grätzer,

General

Lattice Theory

, 2nd Edition, Birkhäuser (1998).

2

R. Sikorski,

Boolean Algebras

, 2nd Edition, Springer-Verlag, New York (1964).

Title

Boolean lattice

Canonical name

BooleanLattice

Date of creation

2013-03-22 12:27:20

Last modified on

2013-03-22 12:27:20

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

19

Author

mathcam (2727)

Entry type

Definition

Classification

msc 06E05

Classification

msc 03G05

Classification

msc 06B20

Classification

msc 03G10

Classification

msc 06E20

Synonym

Boolean algebra

Related topic

BooleanRing

Generated on Fri Feb 9 10:02:31 2018 by

LaTeXML
Boolean prime ideal theorem

Boolean prime ideal theorem

Let

A

be a

Boolean algebra

. Recall that an ideal

I

of

A

if it is

closed under

∨

, and for any

a

∈

I

and

b

∈

A

,

a

∧

b

∈

I

.

I

is proper if

I

≠

A

and non-trivial if

I

≠

(

0

)

, and

I

is prime if it is proper, and, given

a

∧

b

∈

I

, either

a

∈

I

or

b

∈

I

.

Theorem 1

(Boolean prime ideal theorem).

Every Boolean algebra contains a

prime ideal

.

Proof.

Let

A

be a Boolean algebra. If

A

is

trivial

(the two-element

algebra

), then

(

0

)

is the prime ideal we want. Otherwise, pick

a

∈

A

, where

0

≠

a

≠

1

, and let be the trivial ideal. By Birkhoff’s prime ideal theorem for

distributive

lattices,

A

, considered as a

distributive lattice

, has a prime ideal

P

(containing

(

0

)

obviously) such that

a

∉

P

. Then

P

is also a prime ideal of

A

considered as a Boolean algebra.
∎

There are several

equivalent

versions of the

Boolean prime ideal theorem

, some are listed below:

1.

Every Boolean algebra has a prime ideal.

2.

Every ideal in a Boolean algebra can be enlarged to a prime ideal.

3.

Given a set

S

in a Boolean algebra

A

, and an ideal

I

disjoint

from

S

, then there is a prime ideal

P

containing

I

and disjoint from

S

.

4.

An ideal and a filter in a Boolean algebra, disjoint from one another, can be enlarged to an ideal and a filter that are

complement

(as sets) of one another.

Remarks

.

1.

Because the Boolean prime ideal theorem has been extensively studied, it is often abbreviated in the literature as BPI. Since the prime ideal theorem for distributive lattices uses the

axiom of choice

, ZF+AC implies BPI. However, there are models of ZF+BPI where AC fails.

2.

It can be shown (see John Bell’s online article

http://plato.stanford.edu/entries/axiom-choice/

here) that BPI is equivalent, under ZF, to some of the well known

theorems

in mathematics:

–

Tychonoff

’s theorem for

Hausdorff spaces

: the

product

of

compact

Hausdorff spaces is compact under the product topology,

–

the

Stone representation theorem

,

–

the

compactness theorem

(

http://planetmath.org/CompactnessTheoremForFirstOrderLogic

) for

first order logic

, and

–

the

completeness theorem

for first order logic.

References

1

G. Grätzer,

General

Lattice Theory

, 2nd Edition, Birkhäuser (1998).

2

T. J. Jech,

The Axiom of Choice

, North-Holland Pub. Co., Amsterdam, (1973).

3

R. Sikorski,

Boolean Algebras

, 2nd Edition, Springer-Verlag, New York (1964).

Title

Boolean prime ideal theorem

Canonical name

BooleanPrimeIdealTheorem

Date of creation

2013-03-22 18:45:52

Last modified on

2013-03-22 18:45:52

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 06E05

Classification

msc 03G05

Classification

msc 03E25

Synonym

BPI

Generated on Fri Feb 9 10:02:40 2018 by

LaTeXML
derived Boolean operations

derived Boolean operations

Recall that a

Boolean algebra

is an

algebraic system

A

consisting of five

operations

:

1.

two

binary operations

: the meet

∧

and the join

∨

,

2.

one unary operation: the

complementation

′

, and

3.

two nullary operations (

constants

):

0

and

1

.

From these operations, define the following “derived” operations (on

A

): for

a

,

b

∈

A

1.

(

subtraction

)

a

-

b

:=

a

∧

b

′

,

2.

(

symmetric difference

or

addition

)

a

⁢

Δ

⁢

b

(or

a

+

b

)

:=

(

a

-

b

)

∨

(

b

-

a

)

,

3.

(

conditional

)

a

→

b

:=

(

a

-

b

)

′

,

4.

(

biconditional

)

a

↔

b

:=

(

a

→

b

)

∧

(

b

→

a

)

, and

5.

(

Sheffer stroke

)

a

|

b

:=

a

′

∧

b

′

.

Notice that the

operators

→

and

↔

are dual of

-

and

Δ

respectively.

It is evident that these derived operations (and indeed the entire theory of Boolean algebras) owe their

existence

to those operations and

connectives

that are found in logic and

set theory

, as the following table illustrates:

symbol

\

operation

Boolean

Logic

Set

∨

or

∪

join

logical or

union

∧

or

∩

meet

logical and

intersection

′

or

¬

or

∁

complement

logical not

complement

0

bottom

element

falsity

empty set

1

top element

truth

universe

-

or

∖

subtraction

set difference

Δ

or

+

symmetric difference

symmetric difference (

http://planetmath.org/SymmetricDifference

)

→

conditional

implication

↔

biconditional

logical equivalence

|

Sheffer stroke

Sheffer stroke

S

o

m

e

o

f

t

h

e

e

l

e

m

e

n

t

a

r

y

p

r

o

p

e

r

t

i

e

s

o

f

t

h

e

s

e

d

e

r

i

v

e

d

B

o

o

l

e

a

n

o

p

e

r

a

t

o

r

s

a

r

e

:

1.

=

-
a
0

a

and

-
a
a

=

-
0
a

=

-
a
1

=
0

,

2.

(
A
,
+
,
∧
,
0
,
1
)

is a ring (a Boolean ring),

3.

all Boolean operations can be defined in terms of the Sheffer stroke

|

.

T

h

e

p

r

o

o

f

s

o

f

t

h

e

s

e

p

r

o

p

e

r

t

i

e

s

m

i

m

i

c

t

h

e

p

r

o

o

f

s

f

o

r

t

h

e

p

r

o

p

e

r

t

i

e

s

o

f

t

h

e

c

o

r

r

e

s

p

o

n

d

i

n

g

o

p

e

r

a

t

o

r

s

f

o

u

n

d

i

n

n

a

i

v

e

s

e

t

t

h

e

o

r

y

a

n

d

p

r

o

p

o

s

i

t

i

o

n

a

l

l

o

g

i

c

,

s

u

c

h

a

s

t

h

i

s

e

n

t

r

y

(

http://planetmath.org/LogicalConnective

)

.

Title

derived Boolean operations

Canonical name

DerivedBooleanOperations

Date of creation

2013-03-22 17:58:49

Last modified on

2013-03-22 17:58:49

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06E05

Classification

msc 03G05

Classification

msc 06B20

Classification

msc 03G10

Defines

symmetric difference

Defines

conditional

Defines

biconditional

Generated on Fri Feb 9 10:03:12 2018 by

LaTeXML
free Boolean algebra

free Boolean algebra

Let

A

be a

Boolean algebra

and

X

⊆

A

such that

⟨

X

⟩

=

A

. In other words,

X

is a set of

generators

of

A

.

A

is said to be

freely generated

by

X

, or that

X

is a

free set of generators of

A

, if

⟨

X

⟩

=

A

, and every

function

f

from

X

to some Boolean algebra

B

can be extended to a

Boolean algebra homomorphism

g

from

A

to

B

, as illustrated by the

commutative diagram

below:

\xymatrix

⁢

@

⁢

R

-

=

2

⁢

p

⁢

t

⁢

X

⁢

\ar

⁢

[

d

⁢

r

]

f

⁢

\ar

⁢

[

d

⁢

d

]

i

⁢

&

⁢

B

⁢

A

⁢

\ar

⁢

[

u

⁢

r

]

g

where

i

:

X

→

A

is the

inclusion map

. By

extension

of

f

to

g

we mean that

g

⁢

(

x

)

=

f

⁢

(

x

)

for every

x

∈

X

. Any subset

X

⊆

A

containing

0

(or

1

) can never be a

free generating set

for any

subalgebra

of

A

, for any function

f

:

X

→

B

such that

f

⁢

(

0

)

≠

0

can never be extended to a Boolean homomorphism.

A Boolean algebra is said to be

free

if it has a free set of generators. If

A

has

X

as a free set of generators,

A

is said to be

free on

X

. If

A

and

B

are both free on

X

, then

A

and

B

are

isomorphic

. This means that free algebras are uniquely determined by its free generating set, up to

isomorphisms

.

A simple example of a

free Boolean algebra

is the one freely generated by one

element

. Let

X

be a singleton consisting of

a

. Then the set

A

=

{

0

,

a

,

a

′

,

1

}

is a Boolean algebra, with the obvious Boolean operations identified. Every function from

X

to a Boolean algebra

B

singles out an element

b

∈

B

corresponding to

a

. Then the function

g

:

A

→

B

given by

g

⁢

(

a

)

=

b

,

g

⁢

(

a

′

)

=

b

′

,

g

⁢

(

0

)

=

0

, and

g

⁢

(

1

)

=

1

is clearly

Boolean

.

The two-element

algebra

{

0

,

1

}

is also free, its free generating set being

∅

, the

empty set

, since the only function on

∅

is

∅

, and thus can be extended to any function.

In general, if

X

is finite, then the Boolean algebra freely generated by

X

has

cardinality

2

2

|

X

|

, where

|

X

|

is the cardinality of

X

. If

X

is

infinite

, then the cardinality of the Boolean algebra freely generated by

X

is

|

X

|

.

Title

free Boolean algebra

Canonical name

FreeBooleanAlgebra

Date of creation

2013-03-22 18:01:23

Last modified on

2013-03-22 18:01:23

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

12

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06E05

Classification

msc 03G05

Classification

msc 06B20

Classification

msc 03G10

Generated on Fri Feb 9 10:03:32 2018 by

LaTeXML
complete Boolean algebra

complete Boolean algebra

A

Boolean algebra

A

is a

complete Boolean algebra

if for every subset

C

of

A

, the arbitrary join and arbitrary meet of

C

exist.

By de Morgan’s laws, it is easy to see that a Boolean algebra is

complete

iff the arbitrary join of any subset exists iff the arbitrary meet of any subset exists. For a proof of this, see this link (

http://planetmath.org/PropertiesOfArbitraryJoinsAndMeets

).

For an example of a complete Boolean algebra, let

S

be any set. Then the

powerset

P

⁢

(

S

)

with the usual set theoretic

operations

is a complete Boolean algebra.

In a complete Boolean algebra, the

infinite distributive

and

infinite

deMorgan’s laws hold:

•

x

∧

⋁

A

=

⋁

(

x

∧

A

)

and

x

∨

⋀

A

=

⋀

(

x

∨

A

)

•

(

⋁

A

)

*

=

⋀

A

*

and

(

⋀

A

)

*

=

⋁

A

*

, where

A

*

:=

{

a

*

∣

a

∈

A

}

.

In the category of complete Boolean algebras, a

morphism

between two objects is a

Boolean algebra homomorphism

that preserves arbitrary joins (equivalently, arbitrary meets), and is called a

complete Boolean algebra homomorphism

.

Remark

There are infinitely many algebras between Boolean algebras and complete Boolean algebras. Let

κ

be a

cardinal

. A Boolean algebra

A

is said to be

κ

-complete if for every subset

C

of

A

with

|

C

|

≤

κ

,

⋁

C

(and equivalently

⋀

C

) exists. A

κ

-complete Boolean algebra is usually called a

κ

-algebra. If

κ

=

ℵ

0

, the first

aleph number

, then it is called a

countably complete Boolean algebra

.

Any complete Boolean algebra is

κ

-complete, and any

κ

-complete is

λ

-complete for any

λ

≤

κ

. An example of a

κ

-complete algebra that is not complete, take a set

S

with

κ

<

|

S

|

, then the

collection

A

⊆

P

⁢

(

S

)

consisting of any subset

T

such that either

|

T

|

≤

κ

or

|

S

-

T

|

≤

κ

is

κ

-complete but not complete.

A Boolean algebra homomorphism

f

between two

κ

-algebras

A

,

B

is said to be

κ

-complete if

f

⁢

(

⋁

{

a

∣

a

∈

C

}

)

=

⋁

{

f

⁢

(

a

)

∣

a

∈

C

}

for any

C

⊆

A

with

|

C

|

≤

κ

.

Title

complete Boolean algebra

Canonical name

CompleteBooleanAlgebra

Date of creation

2013-03-22 18:01:09

Last modified on

2013-03-22 18:01:09

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

11

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06E10

Related topic

CompleteLattice

Defines

κ

-complete Boolean algebra

Defines

countably complete Boolean algebra

Generated on Fri Feb 9 10:03:42 2018 by

LaTeXML
homeomorphism between Boolean spaces

homeomorphism between Boolean spaces

In this entry, we derive a test for deciding when a

bijection

between two

Boolean spaces

is a

homeomorphism

.

We start with two general remarks.

Lemma 1

.

If

Y

is

zero-dimensional

, then

f

:

X

→

Y

is

continuous

provided that

f

-

1

⁢

(

U

)

is open for every

clopen set

U

in

Y

.

Proof.

Since

Y

is zero-dimensional,

Y

has a basis of clopen sets. To check the

continuity

of

f

, it is enough to check that

f

-

1

⁢

(

U

)

is open for each member of the basis, which is true by

assumption

. Hence

f

is continuous.
∎

Lemma 2

.

If

X

is

compact

and

Y

is

Hausdorff

, and

f

is a bijection, then

f

is a homeomorphism iff it is continuous.

Proof.

One direction is

obvious

. We want to show that

f

-

1

is continuous, or equivalently, for any

closed set

U

in

X

,

f

⁢

(

U

)

is closed in

Y

. Since

X

is compact,

U

is compact, and therefore

f

⁢

(

U

)

is compact since

f

is continuous. But

Y

is Hausdorff, so

f

⁢

(

U

)

is closed.
∎

Proposition 1

.

If

X

,

Y

are Boolean spaces, then a bijection

f

:

X

→

Y

is homeomorphism iff it maps clopen sets to clopen sets.

Proof.

Once more, one direction is clear. Now, suppose

f

maps clopen sets to clopen sets. Since

X

is zero-dimensional,

f

-

1

:

Y

→

X

is continuous by the first

proposition

. Since

Y

is compact and

X

Hausdorff,

f

-

1

is a homeomorphism by the second proposition.
∎

Title

homeomorphism between Boolean spaces

Canonical

name

HomeomorphismBetweenBooleanSpaces

Date of creation

2013-03-22 19:09:04

Last modified on

2013-03-22 19:09:04

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

4

Author

CWoo (3771)

Entry type

Result

Classification

msc 06E15

Classification

msc 06B30

Related topic

DualOfStoneRepresentationTheorem

Generated on Fri Feb 9 10:04:12 2018 by

LaTeXML
representing a Boolean algebra by field of sets

representing a Boolean algebra by field of sets

In this entry, we show that every

Boolean algebra

is

isomorphic

to a

field of sets

, originally noted by Stone in 1936. The bulk of the proof has actually been carried out in this entry (

http://planetmath.org/RepresentingADistributiveLatticeByRingOfSets

), which we briefly state:

if

L

is a

distributive lattice

, and

X

the set of all

prime ideals

of

L

, then the map

F

:

L

→

P

⁢

(

X

)

defined by

F

⁢

(

a

)

=

{

P

∣

a

∉

P

}

is an

embedding

.

Now, if

L

is a Boolean lattice, then every

element

a

∈

L

has a

complement

a

′

∈

L

.

a

′

is in fact uniquely determined by

a

.

Proposition 1

.

The embedding

F

above preserves

′

in the following sense:

F

⁢

(

a

′

)

=

X

-

F

⁢

(

a

)

.

Proof.

P

∈

F

⁢

(

a

′

)

iff

a

′

∉

P

iff

a

∈

P

iff

P

∉

F

⁢

(

a

)

iff

P

∈

X

-

F

⁢

(

a

)

. ∎

Theorem 1

.

Every Boolean algebra is isomorphic to a field of sets.

Proof.

From what has been discussed so far,

F

is a Boolean algebra

isomorphism

between

L

and

F

⁢

(

L

)

, which is a ring of sets first of all, and a field of sets, because

X

-

F

⁢

(

a

)

=

F

⁢

(

a

′

)

. ∎

Remark

. There are at least two other ways to characterize a Boolean algebra as a field of sets: let

L

be a Boolean algebra:

•

Every prime ideal is the kernel of a

homomorphism

into

𝟐

:=

{

0

,

1

}

, and vice versa. So for an element

a

to be not in a prime ideal

P

is the same as saying that

ϕ

⁢

(

a

)

=

1

for some homomorphism

ϕ

:

L

→

𝟐

. If we take

Y

to be the set of all homomorphisms from

L

to

𝟐

, and define

G

:

L

→

P

⁢

(

Y

)

by

G

⁢

(

a

)

=

{

ϕ

∣

ϕ

⁢

(

a

)

=

1

}

, then it is easy to see that

G

is an embedding of

L

into

P

⁢

(

Y

)

.

•

Every prime ideal is a

maximal ideal

, and vice versa. Furthermore,

P

is maximal iff

P

′

is an

ultrafilter

. So if we define

Z

to be the set of all ultrafilters of

L

, and set

H

:

L

→

P

⁢

(

Z

)

by

H

⁢

(

a

)

=

{

U

∣

a

∈

U

}

, then it is easy to see that

H

is an embedding of

L

into

P

⁢

(

Z

)

.

If we appropriately topologize the sets

X

,

Y

, or

Z

, then we have the content of the

Stone representation theorem

.

Title

representing a Boolean algebra by field of sets

Canonical name

RepresentingABooleanAlgebraByFieldOfSets

Date of creation

2013-03-22 19:08:27

Last modified on

2013-03-22 19:08:27

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

13

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 06E20

Classification

msc 06E05

Classification

msc 03G05

Classification

msc 06B20

Classification

msc 03G10

Related topic

FieldOfSets

Related topic

RepresentingADistributiveLatticeByRingOfSets

Related topic

LatticeHomomorphism

Related topic

RepresentingACompleteAtomicBooleanAlgebraByPowerSet

Related topic

StoneRepresentationTheorem

Related topic

MHStonesRepresentationTheorem

Generated on Fri Feb 9 10:04:21 2018 by

LaTeXML
ring hierarchy with links?

ring hierarchy with links?

This isn’t actually a useful

diagram

; see ring hierarchy (

http://planetmath.org/RingHierarchy

) for that. This is just an attempt to replicate it with links in the diagram; it isn’t working well so far. (It does work in the page images view though.)

Ring (unital) (

http://planetmath.org/Ring

)

Commutative ring

Noetherian Ring

(

http://planetmath.org/

Noetherian

)

Local Ring

Title

ring hierarchy with links?

Canonical name

RingHierarchyWithLinks

Date of creation

2013-03-22 18:11:17

Last modified on

2013-03-22 18:11:17

Owner

SamB (21010)

Last modified by

SamB (21010)

Numerical id

9

Author

SamB (21010)

Entry type

Example

Classification

msc 06E20

Generated on Fri Feb 9 10:04:25 2018 by

LaTeXML