aarjaneiro/planetmath huggingface dataset:
example of a non-lattice homomorphism

example of a non-lattice homomorphism

Consider the

Hasse diagram

of the

lattice of subgroups

of the

quaternion group

of order

8

,

Q

8

. [The use of

Q

8

is only for a concrete realization of the

lattice

.]

{xy}

<

5

m

m

,

0

m

m

>

:

<

0

m

m

,

5

m

m

>

:

:

(

0

,

3

)

+

*

Q

8

=

"

Q

8

"

;

(

-

2

,

2

)

+

*

⟨

i

⟩

=

"

i

"

;

(

0

,

2

)

+

*

⟨

j

⟩

=

"

j

"

;

(

2

,

2

)

+

*

⟨

k

⟩

=

"

k

"

;

(

0

,

1

)

+

*

⟨

-

1

⟩

=

"

-

1

"

;

(

0

,

0

)

+

*

⟨

1

⟩

=

"

1

"

;

"

1

"

;

"

-

1

"

*

*

@

-

;

"

-

1

"

;

"

i

"

*

*

@

-

;

"

-

1

"

;

"

j

"

*

*

@

-

;

"

-

1

"

;

"

k

"

*

*

@

-

;

"

i

"

;

"

Q

8

"

*

*

@

-

;

"

j

"

;

"

Q

8

"

*

*

@

-

;

"

k

"

;

"

Q

8

"

*

*

@

-

;

To establish an

order-preserving map

which is not a

lattice isomorphism

one can simply “skip”

⟨

-

1

⟩

, which we display graphically as:

{xy}

<

5

m

m

,

0

m

m

>

:

<

0

m

m

,

5

m

m

>

:

:

(

-

3

,

3

)

+

*

Q

8

=

"

Q

81

"

;

(

-

5

,

2

)

+

*

⟨

i

⟩

=

"

i

1

"

;

(

-

3

,

2

)

+

*

⟨

j

⟩

=

"

j

1

"

;

(

-

1

,

2

)

+

*

⟨

k

⟩

=

"

k

1

"

;

(

-

3

,

1

)

+

*

⟨

-

1

⟩

=

"

-

11

"

;

(

-

3

,

0

)

+

*

⟨

1

⟩

=

"

11

"

;

(

3

,

2.5

)

+

*

Q

8

=

"

Q

82

"

;

(

1

,

1.5

)

+

*

⟨

i

⟩

=

"

i

2

"

;

(

3

,

1.5

)

+

*

⟨

j

⟩

=

"

j

2

"

;

(

5

,

1.5

)

+

*

⟨

k

⟩

=

"

k

2

"

;

(

3

,

0.5

)

+

*

⟨

-

1

⟩

=

"

-

12

"

;

(

3

,

-

0.5

)

+

*

⟨

1

⟩

=

"

12

"

;

"

11

"

;

"

-

11

"

*

*

@

-

;

"

-

11

"

;

"

i

1

"

*

*

@

-

;

"

-

11

"

;

"

j

1

"

*

*

@

-

;

"

-

11

"

;

"

k

1

"

*

*

@

-

;

"

i

1

"

;

"

Q

81

"

*

*

@

-

;

"

j

1

"

;

"

Q

81

"

*

*

@

-

;

"

k

1

"

;

"

Q

81

"

*

*

@

-

;

"

12

"

;

"

-

12

"

*

*

@

-

;

"

-

12

"

;

"

i

2

"

*

*

@

-

;

"

-

12

"

;

"

j

2

"

*

*

@

-

;

"

-

12

"

;

"

k

2

"

*

*

@

-

;

"

i

2

"

;

"

Q

82

"

*

*

@

-

;

"

j

2

"

;

"

Q

82

"

*

*

@

-

;

"

k

2

"

;

"

Q

82

"

*

*

@

-

;

"

Q

81

"

;

"

Q

82

"

*

*

@

.

.

;

"

i

1

"

;

"

i

2

"

*

*

@

.

.

;

"

j

1

"

;

"

j

2

"

*

*

@

.

.

;

"

k

1

"

;

"

k

2

"

*

*

@

.

.

;

"

-

11

"

;

"

12

"

*

*

@

.

.

;

"

11

"

;

"

12

"

*

*

@

.

.

;

Since containment is still preserved the map is order-preserving. However, the

intersection

(meet) of

⟨

i

⟩

and

⟨

j

⟩

, which is

⟨

-

1

⟩

, is not perserved under this
map. Thus it is not a

lattice homomorphism

.

Title

example of a non-lattice

homomorphism

Canonical name

ExampleOfANonlatticeHomomorphism

Date of creation

2013-03-22 16:58:31

Last modified on

2013-03-22 16:58:31

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

8

Author

Algeboy (12884)

Entry type

Example

Classification

msc 06B23

Generated on Fri Feb 9 09:56:45 2018 by

LaTeXML
MacNeille completion

MacNeille completion

In a first course on

real analysis

, one is generally introduced to the

concept

of a

Dedekind cut

. It is a way of constructing the set of real numbers from the

rationals

. This is a process commonly known as the

completion

of the rationals. Three key features of this completion are:

•

the rationals can be embedded in its completion (the reals)

•

every subset with an

upper bound

has a least upper bound

•

every subset with a

lower bound

has a

greatest lower bound

If we extend the reals by adjoining

+

∞

and

-

∞

and define the appropriate

ordering relations

on this new extended set (the

extended real numbers

), then it is a set where every subset has a least upper bound and a greatest lower bound.

When we deal with the rationals and the reals (and extended reals), we are working with

linearly ordered sets

. So the next question is: can the procedure of a completion be generalized to an arbitrary poset? In other words, if

P

is a poset ordered by

≤

, does there exist another poset

Q

ordered by

≤

Q

such that

1.

P

can be embedded in

Q

as a poset (so that

≤

is

compatible

with

≤

Q

), and

2.

every subset of

Q

has both a least upper bound and a greatest lower bound

In 1937, MacNeille answered this question in the affirmative by the following construction:

Given a poset

P

with order

≤

, define for every subset

A

of

P

, two subsets of

P

as follows:

A

u

=

{

p

∈

P

∣

a

≤

p

⁢

for all

⁢

a

∈

A

}

⁢

and

⁢

A

ℓ

=

{

q

∈

P

∣

q

≤

a

⁢

for all

⁢

a

∈

A

}

.

Then

M

⁢

(

P

)

:=

{

A

∈

2

P

∣

(

A

u

)

ℓ

=

A

}

ordered by the usual

set inclusion

is a poset satisfying conditions (1) and (2) above.

This is known as the

MacNeille completion

M

⁢

(

P

)

of a poset

P

. In

M

⁢

(

P

)

, since lub and glb exist for any subset,

M

⁢

(

P

)

is a

complete lattice

. So this process can be readily applied to any

lattice

, if we define a completion of a lattice to follow the two conditions above.

References

1

H. M. MacNeille,

Partially Ordered Sets

. Trans. Amer. Math. Soc. 42 (1937), pp 416-460

2

B. A. Davey, H. A. Priestley,

Introduction

to Lattices and Order

, 2nd edition, Cambridge (2003)

Title

MacNeille completion

Canonical name

MacNeilleCompletion

Date of creation

2013-03-22 16:05:27

Last modified on

2013-03-22 16:05:27

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06B23

Synonym

Dedekind-MacNeille completion

Synonym

normal completion

Related topic

DedekindCuts

Generated on Fri Feb 9 09:56:54 2018 by

LaTeXML
Stone space

Stone space

A

Stone space

, also called a

Boolean space

, is a

topological space

that is

zero-dimensional

,

T

0

(

http://planetmath.org/T0Space

) and

compact

. Equivalently, a Stone space is a

totally disconnected

compact

Hausdorff space

.

Given a stone space

X

, one may

associate

a

Boolean algebra

X

*

by taking the set of all of its

clopen sets

. The set theoretic

operations

of

intersection

, union, and

complement

makes

X

*

a Boolean algebra.

X

*

is known as the

dual algebra

of

X

.

The significance of Stone spaces stems from

Stone duality

:
a pervasive equivalence between the algebraic notions and

theorems

of Boolean algebras on one hand, and the topological notions and
theorems of Stone spaces on the other. This equivalence comprises
the content and

consequences

of M. H. Stone’s representation
theorem.

There is a

bijective

correspondence between the following

1.

The class consisting of all Boolean spaces

2.

The class consisting of all Boolean algebras

3.

The class consisting of all

Boolean rings

4.

The class consisting of all prime spectra of

von Neumann regular rings

In fact, viewing each class as a

category

equipped with the appropriate class of

morphisms

, the categories are

naturally equivalent

between one another.

More to come: partial proofs, constructions,

categorical equivalence

between the first three items due to Stone’s Representation Theorem and

references

…

.

References

1

Paul R. Halmos,

Lectures on Boolean Algebras

, D. Van Nostrand Company, Inc., 1963.

2

Peter T. Johnstone,

Stone Spaces

, Cambridge University Press, 1982.

Title

Stone space

Canonical name

StoneSpace

Date of creation

2013-03-22 13:24:23

Last modified on

2013-03-22 13:24:23

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

19

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06B30

Classification

msc 06E15

Synonym

Boolean space

Related topic

DualityInMathematics

Related topic

DualOfStoneRepresentationTheorem

Defines

dual algebra

Generated on Fri Feb 9 09:57:06 2018 by

LaTeXML
continuous poset

continuous poset

A poset

P

is said to be

continuous

if for every

a

∈

P

1.

the set

wb

⁡

(

a

)

=

{

u

∈

P

∣

u

≪

a

}

is a

directed set

,

2.

⋁

wb

⁡

(

a

)

exists, and

3.

a

=

⋁

wb

⁡

(

a

)

.

In the first condition,

≪

indicates the way below relation on

P

. It is true that in

any

poset, if

b

:=

⋁

wb

⁡

(

a

)

exists, then

b

≤

a

. So for a poset to be continuous, we require that

a

≤

b

.

A

continuous lattice

is a

complete lattice

whose underlying poset is continuous. Note that if

P

is a complete lattice, condition 1 above is automatically satisfied: suppose

u

,

v

≪

a

and

D

⊆

P

with

a

≤

⋁

D

, then there are finite subsets

F

,

G

of

D

with

u

≤

⋁

F

and

v

≤

⋁

G

. Then

H

:=

F

∪

G

⊆

D

is finite and

u

∨

v

≤

(

⋁

F

)

∨

(

⋁

G

)

=

⋁

H

, or

u

∨

v

≪

a

, implying that

wb

⁡

(

a

)

is directed.

Examples

.

1.

Any finite poset is continuous, and so is any finite

lattice

(since it is

complete

).

2.

A chain is continuous iff it is complete.

3.

The

lattice of ideals

of a ring is continuous.

4.

The set of all

lower semicontinuous

functions

from a fixed

compact

topological space

into the extended real numbers is a continuous lattice.

5.

The set of all closed

convex subsets

of a compact convex subset of

ℝ

n

ordered by reverse

inclusion

is a continuous lattice.

Remarks

.

•

Every

algebraic lattice

is continuous.

•

Every continuous

meet semilattice

is

meet continuous

.

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).

Title

continuous poset

Canonical name

ContinuousPoset

Date of creation

2013-03-22 16:43:18

Last modified on

2013-03-22 16:43:18

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06B35

Defines

continuous lattice

Generated on Fri Feb 9 09:57:15 2018 by

LaTeXML
Birkhoff Recurrence Theorem

Birkhoff Recurrence Theorem

Let

T

:

X

→

X

be a

continuous

tranformation in a

compact

metric space

X

. Then, there exists some point

x

∈

X

that is recurrent to

T

, that is, there exists a

sequence

(

n

k

)

k

such that

T

n

k

⁢

(

x

)

→

x

when

k

→

∞

.

Several proofs of this

theorem

are available. It may be obtained from topological

arguments

together with Zorn’s lemma. It is also a

consequence

of Krylov-Bogolyubov theorem, or

existence

of

invariant

probability measures

theorem, which asserts that every continuous transformation in a compact metric space admits an invariant probability measure, and an application of PoincarÃ© Recurrence theorem to that invariant probability measure yields Birkhoff Recurrence theorem.

There is also a

generalization

of Birkhoff recurrence theorem for multiple commuting

transformations

, known as Birkhoff Multiple Recurrence theorem.

Title

Birkhoff Recurrence Theorem

Canonical name

BirkhoffRecurrenceTheorem

Date of creation

2015-03-20 0:56:48

Last modified on

2015-03-20 0:56:48

Owner

Filipe (28191)

Last modified by

Filipe (28191)

Numerical id

2

Author

Filipe (28191)

Entry type

Theorem

Related topic

PoincarÃ© Recurrence Theorem

Generated on Fri Feb 9 09:58:07 2018 by

LaTeXML
Carmichael numbers - 561 (contd)

Carmichael numbers - 561 (contd)

Any

natural number

multiple

of (12 +i), (22 + i), (34 + i) and
(56+ i) is a valid base for pseudoprimality of 561 in
Z(i). 40 is the order of these bases (mod 561 ).

Title

Carmichael numbers

- 561 (contd)

Canonical name

CarmichaelNumbers561contd

Date of creation

2015-07-13 10:29:34

Last modified on

2015-07-13 10:29:34

Owner

akdevaraj (13230)

Last modified by

akdevaraj (13230)

Numerical id

1

Author

akdevaraj (13230)

Entry type

Definition

Generated on Fri Feb 9 09:58:09 2018 by

LaTeXML
Carmichal number 561

Carmichal number 561

561 is the smallest

Carmichael number

in the

ring of
integers

. However it is only a

pseudoprime

in the ring
of

Gaussian integers

. We can use pari to find
the bases to which 561 is a pseudoprime. I
found it is pseudoprime to bases 12 + i, 22 +i,
33 + i and 44 + i. Needless to say it is a
pseudoprime to other bases too. More on this later.

Title

Carmichal number 561

Canonical name

CarmichalNumber561

Date of creation

2015-07-12 6:12:21

Last modified on

2015-07-12 6:12:21

Owner

akdevaraj (13230)

Last modified by

akdevaraj (13230)

Numerical id

1

Author

akdevaraj (13230)

Entry type

Definition

Generated on Fri Feb 9 09:58:10 2018 by

LaTeXML
Carmichal number 561

Carmichal number 561

561 is the smallest

Carmichael number

in the

ring of
integers

. However it is only a

pseudoprime

in the ring
of

Gaussian integers

. We can use pari to find
the bases to which 561 is a pseudoprime. I
found it is pseudoprime to bases 12 + i, 22 +i,
34 + i and 56 + i. Needless to say it is a
pseudoprime to other bases too. More on this later.

Title

Carmichal number 561

Canonical name

CarmichalNumber5611

Date of creation

2015-07-13 4:36:19

Last modified on

2015-07-13 4:36:19

Owner

akdevaraj (13230)

Last modified by

akdevaraj (13230)

Numerical id

2

Author

akdevaraj (13230)

Entry type

Definition

Generated on Fri Feb 9 09:58:12 2018 by

LaTeXML
Carmichal number 561

Carmichal number 561

561 is the smallest

Carmichael number

in the

ring of
integers

. However it is only a

pseudoprime

in the ring
of

Gaussian integers

. We can use pari to find
the bases to which 561 is a pseudoprime. I
found it is pseudoprime to bases 12 + i, 22 +i,
33 + i and 44 + i. Needless to say it is a
pseudoprime to other bases too. More on this later.

Title

Carmichal number 561

Canonical name

CarmichalNumber561

Date of creation

2015-07-12 6:12:19

Last modified on

2015-07-12 6:12:19

Owner

akdevaraj (13230)

Last modified by

akdevaraj (13230)

Numerical id

1

Author

akdevaraj (13230)

Entry type

Definition

Generated on Fri Feb 9 09:58:13 2018 by

LaTeXML
Carmichal number 561

Carmichal number 561

561 is the smallest

Carmichael number

in the

ring of
integers

. However it is only a

pseudoprime

in the ring
of

Gaussian integers

. We can use pari to find
the bases to which 561 is a pseudoprime. I
found it is pseudoprime to bases 12 + i, 22 +i,
33 + i and 44 + i. Needless to say it is a
pseudoprime to other bases too. More on this later.

Title

Carmichal number 561

Canonical name

CarmichalNumber561

Date of creation

2015-07-12 6:12:20

Last modified on

2015-07-12 6:12:20

Owner

akdevaraj (13230)

Last modified by

akdevaraj (13230)

Numerical id

1

Author

akdevaraj (13230)

Entry type

Definition

Generated on Fri Feb 9 09:58:15 2018 by

LaTeXML
convergence in the mean

convergence in the mean

Let

b

n

:=

a

1

+

a

2

+

…

+

a

n

n

(

n

=

1

,

2

,

3

,

…

)

be the

arithmetic mean

of the numbers

a

1

,

a

2

,

…

,

a

n

.
The

sequence

a

1

,

a

2

,

a

3

,

…

(1)

is said to

converge

in the mean

(

http://planetmath.org/ConvergenceInTheMean

) iff the
sequence

b

1

,

b

2

,

b

3

,

…

(2)

converges.

On has the

Theorem.

If the sequence (1) is

convergent

having the limit

A

, then also the sequence
(2) converges to the limit

A

.  Thus, a

convergent sequence

is always convergent in the mean.

Proof.

Let

ε

be an arbitrary positive number.  We may write

|

A

-

b

n

|

=

|

A

-

1

n

⁢

(

a

1

+

…

+

a

k

)

-

1

n

⁢

(

a

k

+

1

+

…

+

a

n

)

|

=

|

1

n

⁢

[

(

A

-

a

1

)

+

…

+

(

A

-

a

k

)

]

+

1

n

⁢

[

(

A

-

a

k

+

1

)

+

…

+

(

A

-

a

n

)

]

|

≦

|

(

A

-

a

1

)

+

…

+

(

A

-

a

k

)

|

n

+

|

A

-

a

k

+

1

|

+

…

+

|

A

-

a

n

|

n

.

The supposition implies that there is a

positive integer

k

such that

|

A

-

a

i

|

<

ε

2

for all

⁢

i

>

k

.

Let’s fix the

integer

k

.  Choose the number

l

so great that

|

(

A

-

a

1

)

+

…

+

(

A

-

a

k

)

|

n

<

ε

2

for

⁢

n

>

l

.

Let now

n

>

max

⁡

{

k

,

l

}

.  The three above

inequalities

yield

|

A

-

b

n

|

<

ε

2

+

1

n

⁢

(

n

-

k

)

⁢

ε

2

<

ε

2

+

ε

2

=

ε

,

whence we have

lim

n

→

∞

⁡

b

n

=

A

.

Note.

The converse (

http://planetmath.org/Converse

) of the theorem is not
true.  For example, if

a

n

:=

1

+

(

-

1

)

n

2

i.e. if the sequence (1) has the form

0

,

1

,

0

,

1

,

0

,

1

,

…

,

then it is divergent but converges in the mean to the limit

1

2

; the corresponding sequence (2) is

0

,

1

2

,

1

3

,

2

4

,

2

5

,

3

6

,

3

7

,

4

8

,

4

9

,

…

Title

convergence

in the mean

Canonical name

ConvergenceInTheMean

Date of creation

2015-04-08 7:29:35

Last modified on

2015-04-08 7:29:35

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Definition

Generated on Fri Feb 9 09:58:18 2018 by

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definition of vector space needs no commutativity

definition of vector space needs no commutativity

In the

definition

of

vector space

(

http://planetmath.org/VectorSpace

) one
usually lists the needed

properties

of the vectoral

addition

and the multiplication of vectors by scalars as eight axioms,
one of them the

commutative law

u

+

v

=

v

+

u

.

The latter is however not necessary, because it may be proved
to be a

consequence

of the other seven axioms.  The proof can
be based on the fact that in defining the group (

http://planetmath.org/Group

),
it suffices to

postulate

only the

existence

of a

right identity

element

and the

right inverses

of the elements (see the article
“redundancy of two-sidedness in definition of group (

http://planetmath.org/RedundancyOfTwoSidednessInDefinitionOfGroup

)”).

Now, suppose the

validity

of the seven other axioms (

http://planetmath.org/VectorSpace

), but not necessarily the above commutative law of
addition.  We will show that the commutative law is in force.

We need the

identity

(

-

1

)

⁢

v

=

-

v

which is easily justified
(we have

0

→

=

0

⁢

v

=

(

1

+

(

-

1

)

)

⁢

v

=

…

).  Then we can
calculate as follows:

v

+

u

=

(

v

+

u

)

+

0

→

=

(

v

+

u

)

+

[

-

(

u

+

v

)

+

(

u

+

v

)

]

=

[

(

v

+

u

)

+

(

-

(

u

+

v

)

)

]

+

(

u

+

v

)

=

[

(

v

+

u

)

+

(

-

1

)

⁢

(

u

+

v

)

]

+

(

u

+

v

)

=

[

(

v

+

u

)

+

(

(

-

1

)

⁢

u

+

(

-

1

)

⁢

v

)

]

+

(

u

+

v

)

=

[

(

(

v

+

u

)

+

(

-

u

)

)

+

(

-

v

)

]

+

(

u

+

v

)

=

[

(

v

+

(

u

+

(

-

u

)

)

)

+

(

-

v

)

]

+

(

u

+

v

)

=

[

(

v

+

0

→

)

+

(

-

v

)

]

+

(

u

+

v

)

=

[

v

+

(

-

v

)

]

+

(

u

+

v

)

=

0

→

+

(

u

+

v

)

=

u

+

v

Q.E.D.

This proof by

Y. Chemiavsky

and

A. Mouftakhov

is
found in the 2012 March issue of

The American Mathematical
Monthly

.

Title

definition of vector space needs no commutativity

Canonical

name

DefinitionOfVectorSpaceNeedsNoCommutativity

Date of creation

2015-01-25 12:26:14

Last modified on

2015-01-25 12:26:14

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

5

Author

pahio (2872)

Entry type

Feature

Generated on Fri Feb 9 09:58:21 2018 by

LaTeXML
difference of lattice elements

difference of lattice elements

Let

𝔄

is a

lattice

with least

element

0

.

Let

a

,

b

∈

𝔄

. A

of

a

and

b

is an element

c

∈

𝔄

that

b

∩

c

=

0

and

a

∪

b

=

b

∪

c

. When there is only one

difference

of

a

and

b

, it is denoted

a

∖

b

.

One immediate

property

is:

0

is the unique difference of any element

a

and itself (

a

∖

a

=

0

). For if

c

is such a difference, then

a

∩

c

=

0

and

a

=

a

∪

c

. So

c

≤

a

by the second equation, and hence that

c

=

a

∩

c

=

0

by the first equation.

For arbitrary lattices

of two given elements do not necessarily exist. For some lattices there may be more than one difference of two given elements.

For a

distributive lattice

with bottom element

0

, the difference of two elements, if it exists, must be unique. To see this, let

c

and

d

be two differences of

a

and

b

. Then

•

b

∩

c

=

b

∩

d

=

0

, and

•

a

∪

b

=

b

∪

c

=

b

∪

d

.

So

c

=

c

∩

(

b

∪

c

)

=

c

∩

(

b

∪

d

)

=

(

c

∩

b

)

∪

(

c

∩

d

)

=

0

∪

(

c

∩

d

)

=

c

∩

d

. Similarly,

d

=

d

∩

c

. As a result,

c

=

c

∩

d

=

d

∩

c

=

d

.

Title

difference of lattice elements

Canonical

name

DifferenceOfLatticeElements

Date of creation

2013-03-22 17:57:44

Last modified on

2013-03-22 17:57:44

Owner

porton (9363)

Last modified by

porton (9363)

Numerical id

10

Author

porton (9363)

Entry type

Definition

Classification

msc 06B99

Related topic

ComplementedLattice

Related topic

Pseudodifference

Related topic

SectionallyComplementedLattice

Generated on Fri Feb 9 09:58:23 2018 by

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field arising from special relativity

field arising from special relativity

The velocities

u

and

v

of two bodies moving along a line
obey, by the special theory of relativity, the

addition

rule

u

⊕

v

:=

u

+

v

1

+

u

⁢

v

c

2

,

(1)

where

c

is the velocity of light.  As

c

is unreachable for any material body, it
plays for the velocities of the bodies the role of the

infinity

.  These velocities

v

thus

satisfy

always

|

v

|

<

c

.

By (1) we get

c

⊕

c

=

c

,

c

⊕

v

=

c

for

|

v

|

<

c

;  so

c

behaves like the infinity.

One can define the

mapping

(

http://planetmath.org/mapping

)

f

:

ℝ

→

(

-

c

,

c

)

=

S

by setting

f

⁢

(

x

)

:=

c

⁢

tanh

⁡

x

(2)

which is easily seen to be a

bijection

.

Define also the

binary operation

(

http://planetmath.org/binaryoperation

)

⊙

for the numbers (

http://planetmath.org/number

)

u

,

v

of the

open interval

(

http://planetmath.org/

interval

)

(

-

c

,

c

)

by

u

⊙

v

=

c

⁢

tanh

⁡

[

(

artanh

⁢

u

c

)

⁢

(

artanh

⁢

v

c

)

]

.

(3)

Then the system

(

S

,

⊕

,

⊙

)

may be checked to be a ring
and the

bijective

mapping (2) to be
homomorphic (

http://planetmath.org/structurehomomorphism

):

f

⁢

(

x

+

y

)

=

f

⁢

(

x

)

⊕

f

⁢

(

y

)

,

f

⁢

(

x

⁢

y

)

=

f

⁢

(

x

)

⊙

f

⁢

(

y

)

Consequently, the system

(

S

,

⊕

,

⊙

)

, as the

homomorphic image

(

http://planetmath.org/homomorphicimageofgroup

) of
the field

(

ℝ

,

+

,

⋅

)

, also itself is a field.

Baker [1] calls the numbers of the set

S

, i.e.

(

-

c

,

c

)

,
the

Einstein numbers

.

References

1

G. A. Baker, Jr.

: “Einstein numbers”. –

Amer. Math. Monthly

61

(1954), 39–41.

2

H. T. Davis

:

College

algebra

. Prentice-Hall, N.Y. (1940), 351.

3

T. Gregor & J. Haluška

:

Two-dimensional Einstein
numbers and

associativity

.

http://arxiv.org/abs/1309.0660

arXiv (2013)

Title

field arising from special relativity

Canonical name

FieldArisingFromSpecialRelativity

Date of creation

2016-04-20 13:42:53

Last modified on

2016-04-20 13:42:53

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Topic

Generated on Fri Feb 9 09:58:27 2018 by

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