aarjaneiro/planetmath huggingface dataset:
Filtrator

Filtrator

A

filtrator

is a pair

(

𝔄

;

ℨ

)

consisting of a poset

𝔄

(the

base

of the filtrator) and its subset

ℨ

(the

core

of the filtrator). The set

ℨ

is considered as a poset with the induced order.

Having fixed a filtrator and an

a

∈

𝔄

, we define:

up

⁡

a

=

{

X

∈

ℨ

|

X

≥

a

}

down

⁡

a

=

{

X

∈

ℨ

|

X

≤

a

}

.

Probably the most important example of a filtrator is a

primary filtrator

that is the pair

(

𝔉

;

𝔓

)

where

𝔉

is the set of filters on a poset ordered reverse to set-theoretic

inclusion

of filters and

𝔓

is the set of

principal filters

on this poset. For a filter

ℱ

∈

𝔉

we have

up

⁡

ℱ

essentially

equivalent

with the set

ℱ

.

References

1

Victor Porton.

http://www.mathematics21.org/binaries/filters.pdf

Filters on posets and

generalizations

. International Journal of Pure and
Applied Mathematics, 74(1):55–119, 2012.

Title

Filtrator

Canonical name

Filtrator

Date of creation

2013-03-22 19:31:25

Last modified on

2013-03-22 19:31:25

Owner

porton (9363)

Last modified by

porton (9363)

Numerical id

6

Author

porton (9363)

Entry type

Definition

Classification

msc 06B99

Classification

msc 06A06

Classification

msc 54A20

Related topic

Filter

Related topic

Filter2

Defines

primary filtrator

Generated on Fri Feb 9 09:58:29 2018 by

LaTeXML
join irreducibility

join irreducibility

An element

a

in a

lattice

L

is said to be

join irreducible

iff

a

is not a bottom element, and, whenever

a

=

b

∨

c

, then

a

=

b

or

a

=

c

. Dually,

a

∈

L

is

meet irreducible

iff

a

is not a top element, and, whenever

a

=

b

∧

c

, then

a

=

b

or

a

=

c

. If

a

is both join and meet

irreducible

, then

a

is said to be

irreducible

. Any atom in a lattice is join irreducible.

Example

. In the lattice

diagram

(

Hasse diagram

) below,

\xymatrix

⁢

&

⁢

1

⁢

\ar

⁢

@

-

[

d

]

⁢

&

⁢

&

⁢

a

⁢

\ar

⁢

@

-

[

l

⁢

d

]

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

b

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

&

⁢

&

⁢

c

⁢

\ar

⁢

@

-

[

l

⁢

d

]

⁢

&

⁢

d

⁢

\ar

⁢

@

-

[

d

]

⁢

&

⁢

&

⁢

0

a

is meet irreducible but not join irreducible,

d

is join irreducible but not meet irreducible, while

b

,

c

are irreducible.

From this, we make the observations that in any chain, all the elements except the bottom one are join irreducible. Dually, all the elements except the top one are meet irreducible. An element is join irreducible iff it covers (

http://planetmath.org/CoveringRelation

) at most one other element. An element is meet irreducible iff it is covered by at most one other element.

Remark.

If a lattice satisfies the

descending chain condition

, then every element can be expressed as a join of join irreducible elements. This statement can be dualized: if a lattice satisfies the

ascending chain condition

, then every element is the meet of meet irreducible elements.

References

1

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

Introduction to Lattices and Order

, 2nd Edition, Cambridge (2003)

Title

join irreducibility

Canonical name

JoinIrreducibility

Date of creation

2013-03-22 15:47:29

Last modified on

2013-03-22 15:47:29

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06B99

Synonym

join-irreducible

Synonym

meet-irreducible

Defines

join irreducible

Defines

meet irreducible

Defines

irreducible

Generated on Fri Feb 9 09:58:33 2018 by

LaTeXML
lattice interval

lattice interval

Definition

. Let

L

be a

lattice

. A subset

I

of

L

is called a

lattice interval

, or simply an

if there exist

elements

a

,

b

∈

L

such that

I

=

{

t

∈

L

∣

a

≤

t

≤

b

}

:=

[

a

,

b

]

.

The elements

a

,

b

are called the

endpoints

of

I

. Clearly

a

,

b

∈

I

. Also, the endpoints of a lattice interval are unique: if

[

a

,

b

]

=

[

c

,

d

]

, then

a

=

c

and

b

=

d

.

Remarks

.

•

It is easy to see that the name is derived from that of an

interval

on a number line. From this

analogy

, one can easily define lattice intervals without one or both endpoints. Whereas an interval on a number line is

linearly ordered

, a lattice interval in general is not. Nevertheless, a lattice interval

I

of a lattice

L

is a sublattice of

L

.

•

A

bounded lattice

is itself a lattice interval:

[

0

,

1

]

.

•

A

prime interval

is a lattice interval that contains its endpoints and nothing else. In other words, if

[

a

,

b

]

is prime, then any

c

∈

[

a

,

b

]

implies that either

c

=

a

or

c

=

b

. Simply put,

b

covers

a

. If a lattice

L

contains

0

, then for any

a

∈

L

,

[

0

,

a

]

is a prime interval iff

a

is an atom.

•

Since no

operations

of meet and join are used, all of the above discussion can be generalized to define an interval in a poset.

•

Given a lattice

L

, let

ℬ

be the

collection

of all lattice intervals without endpoints, we can form a topolgy on

L

with

ℬ

as the

subbasis

. This does not insure that

∧

and

∨

are

continuous

, so that

L

with this topological structure may not be a

topological lattice

.

•

Locally Finite Lattice

. A lattice that is derived based on the

concept

of lattice interval is that of a locally finite lattice. A lattice

L

is

locally finite

iff every one of its interval is finite. Unless the lattice is finite, a locally finite lattice, if

infinite

, is either topless or bottomless.

Title

lattice interval

Canonical name

LatticeInterval

Date of creation

2013-03-22 15:44:56

Last modified on

2013-03-22 15:44:56

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

11

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06B99

Classification

msc 06A06

Defines

prime interval

Defines

poset interval

Defines

locally finite lattice

Generated on Fri Feb 9 09:58:34 2018 by

LaTeXML
Multiple Recurrence Theorem

Multiple Recurrence Theorem

Let

(

X

,

ℬ

,

μ

)

be a

probability space

, and let

T

i

:

X

→

X

be

measure-preserving transformations

, for

i

between

1

and

q

. Assume that all the

transformations

T

i

commute. If

E

⊂

X

is a

positive measure

set

μ

⁢

(

E

)

>

0

, then, there exists

n

∈

ℕ

such that

μ

⁢

(

E

∩

T

1

-

n

⁢

(

E

)

∩

⋯

∩

T

q

-

n

⁢

(

E

)

)

>

0

In other words there exist a certain time

n

such that the subset of

E

for which all elements return to

E

simultaneously for all transformations

T

i

is a subset of

E

with positive measure.
Observe that the theorem may be applied again to the set

G

=

E

∩

T

1

-

n

⁢

(

E

)

∩

⋯

∩

T

q

-

n

⁢

(

E

)

, obtaining the existence of

m

∈

ℕ

such that

μ

⁢

(

G

∩

T

1

-

m

⁢

(

G

)

∩

⋯

∩

T

q

-

m

⁢

(

G

)

)

>

0

so that

μ

⁢

(

E

∩

T

1

-

(

m

+

n

)

⁢

(

E

)

∩

⋯

∩

T

q

-

(

m

+

n

)

⁢

(

E

)

)

≥

μ

⁢

(

G

∩

T

1

-

m

⁢

(

G

)

∩

⋯

∩

T

q

-

m

⁢

(

G

)

)

>

0

So we may conclude that, when

E

has positive measure, there are

infinite

times for which there is a simultaneous return for a subset of

E

with positive measure.

As a corollary, since the powers

T

,

T

2

⁢

⋯

⁢

T

q

of a transformation

T

commute, we have that, for

E

with positive measure there exists

n

∈

ℕ

such that

μ

⁢

(

E

∩

T

-

n

⁢

(

E

)

∩

⋯

∩

T

-

q

⁢

n

⁢

(

E

)

)

>

0

As a consequence of the multiple recurrence theorem one may prove SzemerÃ©di’s Theorem about arithmetic progressions.

Title

Multiple Recurrence Theorem

Canonical name

MultipleRecurrenceTheorem

Date of creation

2015-03-20 0:29:34

Last modified on

2015-03-20 0:29:34

Owner

Filipe (28191)

Last modified by

Filipe (28191)

Numerical id

1

Author

Filipe (28191)

Entry type

Theorem

Synonym

PoincarÃ© Multiple Recurrence Theorem; FÃ¼rstenberg Recurrence theorem

Related topic

PoincarÃ© Recurrence Theorem

Generated on Fri Feb 9 09:58:48 2018 by

LaTeXML
Nucleus

Nucleus

In order theory, a

nucleus

is a

function

F

on a

meet-semilattice

𝔄

such that (for every

p

in

𝔄

):

1.

p

≤

F

⁢

(

p

)

2.

F

⁢

(

F

⁢

(

p

)

)

=

F

⁢

(

p

)

3.

F

⁢

(

p

∧

q

)

=

F

⁢

(

p

)

∧

F

⁢

(

q

)

Usually, the term

nucleus

is used in frames and locales theory (when the

semilattice

𝔄

is a frame).

1

Some well known results about nuclei

Proposition

If

F

is a nucleus on a frame

𝔄

, then the poset

Fix

⁡

(

F

)

of

fixed points

of

F

, with order inherited from

𝔄

, is also a frame.

Title

Nucleus

Canonical

name

Nucleus

Date of creation

2014-12-18 15:34:14

Last modified on

2014-12-18 15:34:14

Owner

porton (9363)

Last modified by

porton (9363)

Numerical id

1

Author

porton (9363)

Entry type

Definition

Classification

msc 06B99

Generated on Fri Feb 9 09:58:50 2018 by

LaTeXML
proof of Schroeder-Bernstein theorem using Tarski-Knaster theorem

proof of Schroeder-Bernstein theorem using Tarski-Knaster theorem

The

Tarski-Knaster theorem

can be used to give a short, elegant proof of the

Schroeder-Bernstein theorem

.

Proof.

Suppose

f

:

S

→

T

and

g

:

T

→

S

are

injective

. Define a

function

φ

:

P

⁢

(

S

)

→

P

⁢

(

S

)

by

φ

⁢

(

X

)

=

S

∖

g

⁢

(

T

∖

f

⁢

(

X

)

)

.

If

X

⊆

Y

⊆

S

, then

S

∖

g

⁢

(

T

∖

f

⁢

(

X

)

)

⊆

S

∖

g

⁢

(

T

∖

f

⁢

(

Y

)

)

, and so

φ

is

monotone

. Since

P

⁢

(

S

)

is a

complete lattice

, we may apply the Tarski-Knaster theorem to conclude that the set of

fixed points

of

φ

is a complete lattice and thus nonempty.

Let

C

be a fixed point of

φ

. We have

S

∖

C

=

g

⁢

(

T

∖

f

⁢

(

C

)

)

.

Hence

g

|

T

∖

f

⁢

(

C

)

:

T

∖

f

⁢

(

C

)

→

S

∖

C

and

f

|

C

:

C

→

f

⁢

(

C

)

are

bijections

. We can therefore construct the desired bijection

h

:

S

→

T

by defining

h

⁢

(

x

)

=

{

f

⁢

(

x

)

if

⁢

x

∈

C

(

g

|

T

∖

f

⁢

(

C

)

)

-

1

⁢

(

x

)

if

⁢

x

∉

C

.

∎

The usual proof of Schroeder-Bernstein theorem explicitly constructs a fixed point of

φ

.

References

1

Thomas Forster,

Logic,

induction

and sets

, Cambridge University Press, Cambridge, 2003.

2

M. Kolibiar, A. Legéň, T. Šalát, and Š. Znám,

Algebra a príbuzné disciplíny

, Alfa, Bratislava, 1992 (Slovak).

Title

proof of Schroeder-Bernstein theorem using Tarski-Knaster theorem

Canonical name

ProofOfSchroederBernsteinTheoremUsingTarskiKnasterTheorem

Date of creation

2013-03-22 15:30:24

Last modified on

2013-03-22 15:30:24

Owner

kompik (10588)

Last modified by

kompik (10588)

Numerical id

7

Author

kompik (10588)

Entry type

Proof

Classification

msc 06B99

Related topic

SchroederBernsteintheorem

Related topic

TarskiKnastertheorem

Related topic

TarskiKnasterTheorem

Related topic

SchroederBernsteinTheorem

Generated on Fri Feb 9 09:58:52 2018 by

LaTeXML
pseudodifference

pseudodifference

An element

z

of

lattice

L

is

pseudodifference

of

y

and

x

(

y

∖

x

) if

z

is the least element such that

y

≤

x

∪

z

.

Pseudodifference

is denoted either as

∖

or as

-

. Sometimes

pseudodifference

is denoted as

∖

*

.

The definition is borrowed from

http://www.hindawi.com/Getpdf.aspx?doi=10.1155/S0161171280000312

this online article.

Title

pseudodifference

Canonical name

Pseudodifference

Date of creation

2013-03-22 17:58:35

Last modified on

2013-03-22 17:58:35

Owner

porton (9363)

Last modified by

porton (9363)

Numerical id

5

Author

porton (9363)

Entry type

Definition

Classification

msc 06B99

Synonym

pseudo difference

Related topic

Pseudocomplement

Related topic

DifferenceOfLatticeElements

Generated on Fri Feb 9 09:58:54 2018 by

LaTeXML
redundancy of two-sidedness in definition of group

redundancy of two-sidedness in definition of group

In the definition of group, one usually supposes that there is a two-sided

identity element

and that
any

element

has a two-sided

inverse

(cf. group (

http://planetmath.org/Group

)).

The group may also be defined without the two-sidednesses:

A

group

is a pair of a non-empty set

G

and its

associative

binary operation

(

x

,

y

)

↦

x

⁢

y

such that

1) the

operation

has a

right identity

element

e

;

2) any element

x

of

G

has a

right inverse

x

-

1

.

We have to show that the right identity

e

is also a left identity
and that any right inverse is also a

left inverse

.

Let the above

assumptions

on

G

be true.  If

a

-

1

is the right
inverse of an arbitrary element

a

of

G

, the calculation

a

-

1

⁢

a

=

a

-

1

⁢

a

⁢

e

=

a

-

1

⁢

a

⁢

a

-

1

⁢

(

a

-

1

)

-

1

=

a

-

1

⁢

e

⁢

(

a

-

1

)

-

1

=

a

-

1

⁢

(

a

-

1

)

-

1

=

e

shows that it is also the left inverse of

a

.  Using this result,
we then can write

e

⁢

a

=

(

a

⁢

a

-

1

)

⁢

a

=

a

⁢

(

a

-

1

⁢

a

)

=

a

⁢

e

=

a

,

whence

e

is a left identity element, too.

Title

redundancy of two-sidedness in definition of group

Canonical name

RedundancyOfTwosidednessInDefinitionOfGroup

Date of creation

2015-01-20 17:28:03

Last modified on

2015-01-20 17:28:03

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

3

Author

pahio (2872)

Entry type

Definition

Generated on Fri Feb 9 09:58:56 2018 by

LaTeXML
residuated lattice

residuated lattice

A

residuated lattice

is a

lattice

L

with an additional

binary operation

⋅

called

multiplication

, with a

multiplicative identity

e

∈

L

, such that

•

(

L

,

⋅

,

e

)

is a monoid, and

•

for each

x

∈

L

, the left and right multiplications by

x

are

residuated

.

The second condition says: for every

x

,

z

∈

L

, each of the sets

L

⁢

(

x

,

z

)

:=

{

y

∈

L

∣

x

⋅

y

≤

z

}

and

R

⁢

(

x

,

z

)

:=

{

y

∈

L

∣

y

⋅

x

≤

z

}

is a down set, and has a maximum.

Clearly,

max

⁡

L

⁢

(

x

,

z

)

and

max

⁡

R

⁢

(

x

,

z

)

are both unique.

max

⁡

L

⁢

(

x

,

z

)

is called the

right residual

of

z

by

x

, and is commonly denoted by

x

\

z

, while

max

⁡

R

⁢

(

x

,

z

)

is called the

left residual

of

z

by

x

, denoted by

x

/

z

.

Residuated lattices are mostly found in

algebraic structures

associated with a

variety

of

logical systems

. For examples,

Boolean algebras

associated with classical

propositional logic

, and more generally

Heyting algebras

associated with the intuitionistic propositional logic are both residuated, with multiplication the same as the lattice meet

operation

. MV-algebras and BL-algebras associated with

many-valued logics

are further examples of residuated lattices.

Remark

. A residuated lattice is said to be

commutative

if

⋅

is commutative. All of the examples cited above are commutative.

References

1

T.S. Blyth,

Lattices and Ordered Algebraic Structures

, Springer, New York (2005)

2

M. Bergmann,

An

Introduction

to Many-Valued and

Fuzzy Logic

:

Semantic

,

Algebras

, and

Derivation

Systems

, Cambridge University Press (2008)

3

R. P. Dilworth, M. Ward

Residuated Lattices

, Transaction of the American Mathematical Society 45, pp.335-354 (1939)

Title

residuated lattice

Canonical name

ResiduatedLattice

Date of creation

2013-03-22 18:53:41

Last modified on

2013-03-22 18:53:41

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06B99

Defines

left residual

Defines

right residual

Defines

commutative residuated lattice

Generated on Fri Feb 9 09:59:05 2018 by

LaTeXML
special elements in a lattice

special elements in a lattice

Let

L

be a

lattice

and

a

∈

L

is said to be

•

distributive

if

a

∨

(

b

∧

c

)

=

(

a

∨

b

)

∧

(

a

∨

c

)

,

•

standard

if

b

∧

(

a

∨

c

)

=

(

b

∧

a

)

∨

(

b

∧

c

)

, or

•

neutral

if

(

a

∧

b

)

∨

(

b

∧

c

)

∨

(

c

∧

a

)

=

(

a

∨

b

)

∧

(

b

∨

c

)

∧

(

c

∨

a

)

for all

b

,

c

∈

L

. There are also dual notions of the three types mentioned above, simply by exchanging

∨

and

∧

in the definitions. So a

dually distributive

element

a

∈

L

is one where

a

∧

(

b

∨

c

)

=

(

a

∧

b

)

∨

(

a

∧

c

)

for all

b

,

c

∈

L

, and a

dually standard element

is similarly defined. However, a

dually neutral

element is the same as a

neutral element

.

Remarks

For any

a

∈

L

, suppose

P

is the property in

L

such that

a

∈

P

iff

a

∨

b

=

a

∨

c

and

a

∧

b

=

a

∧

c

imply

b

=

c

for all

b

,

c

∈

L

.

•

A standard element is distributive.

Conversely

, a distributive satisfying

P

is standard.

•

A neutral element is distributive (and consequently dually distributive). Conversely, a distributive and dually distributive element that

satisfies

P

is neutral.

References

1

G. Birkhoff

Lattice Theory

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

2

G. Grätzer,

General Lattice Theory

, 2nd Edition, Birkhäuser (1998).

Title

special elements in a lattice

Canonical name

SpecialElementsInALattice

Date of creation

2013-03-22 16:42:29

Last modified on

2013-03-22 16:42:29

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06B99

Defines

distributive element

Defines

standard element

Defines

neutral element

Defines

dually distributive

Defines

dually standard

Generated on Fri Feb 9 09:59:14 2018 by

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modular inequality

modular inequality

In any lattice (

http://planetmath.org/lattice

) the self-dual

modular

inequality

is true:
if

x

≤

z

then

x

∨

(

y

∧

z

)

≤

(

x

∨

y

)

∧

z

.

Proof.

x

≤

x

∨

y

and we are given that

x

≤

z

, so

x

≤

(

x

∨

y

)

∧

z

.
Also,

y

∧

z

≤

y

≤

x

∨

y

and

y

∧

z

≤

z

imply that

y

∧

z

≤

(

x

∨

y

)

∧

z

.
Therefore,

x

∨

(

y

∧

z

)

≤

(

x

∨

y

)

∧

z

.
∎

Title

modular inequality

Canonical name

ModularInequality

Date of creation

2014-02-01 1:48:21

Last modified on

2014-02-01 1:48:21

Owner

ixionid (16766)

Last modified by

ixionid (16766)

Numerical id

10

Author

ixionid (16766)

Entry type

Theorem

Classification

msc 06C05

Related topic

ModularLattice

Related topic

DistributiveInequalities

Defines

modular inequality

Generated on Fri Feb 9 09:59:21 2018 by

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modular lattice

modular lattice

A

lattice

L

is said to be

modular

if

x

∨

(

y

∧

z

)

=

(

x

∨

y

)

∧

z

for all

x

,

y

,

z

∈

L

such that

x

≤

z

.
In fact it is

sufficient

to show that

x

∨

(

y

∧

z

)

≥

(

x

∨

y

)

∧

z

for all

x

,

y

,

z

∈

L

such that

x

≤

z

,
as the reverse

inequality

holds in all lattices (see modular inequality).

There are a number of other

equivalent

conditions for a lattice

L

to be modular:

•

(

x

∧

y

)

∨

(

x

∧

z

)

=

x

∧

(

y

∨

(

x

∧

z

)

)

for all

x

,

y

,

z

∈

L

.

•

(

x

∨

y

)

∧

(

x

∨

z

)

=

x

∨

(

y

∧

(

x

∨

z

)

)

for all

x

,

y

,

z

∈

L

.

•

For all

x

,

y

,

z

∈

L

,
if

x

<

z

then either

x

∧

y

<

z

∧

y

or

x

∨

y

<

z

∨

y

.

The following are examples of modular lattices.

•

All

distributive

lattices (

http://planetmath.org/DistributiveLattice

).

•

The lattice of

normal subgroups

of any group.

•

The lattice of

submodules

of any module (

http://planetmath.org/Module

).
(See modular law.)

A finite lattice

L

is modular
if and only if it
is graded and its

rank function

ρ

satisfies

ρ

⁢

(

x

)

+

ρ

⁢

(

y

)

=

ρ

⁢

(

x

∧

y

)

+

ρ

⁢

(

x

∨

y

)

for all

x

,

y

∈

L

.

Title

modular lattice

Canonical name

ModularLattice

Date of creation

2013-03-22 12:27:26

Last modified on

2013-03-22 12:27:26

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

17

Author

yark (2760)

Entry type

Definition

Classification

msc 06C05

Synonym

Dedekind lattice

Related topic

ModularLaw

Related topic

SemimodularLattice

Related topic

NonmodularSublattice

Related topic

ModularInequality

Defines

modular

Generated on Fri Feb 9 09:59:23 2018 by

LaTeXML
nonmodular sublattice

nonmodular sublattice

Any nonmodular

lattice

L

contains the lattice

N

5

(shown below) as a sublattice.

\xymatrix

⁢

&

⁢

x

∨

y

⁢

\ar

⁢

@

-

[

l

⁢

d

]

⁢

\ar

⁢

@

-

[

r

⁢

d

⁢

r

⁢

d

]

⁢

&

⁢

&

⁢

(

x

∨

y

)

∧

z

⁢

\ar

⁢

@

-

[

d

⁢

d

]

⁢

&

⁢

&

⁢

&

⁢

&

⁢

&

⁢

&

⁢

y

⁢

\ar

⁢

@

-

[

l

⁢

d

⁢

l

⁢

d

]

⁢

x

∨

(

y

∧

z

)

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

&

⁢

&

⁢

&

⁢

&

⁢

y

∧

z

⁢

&

⁢

&

Proof.

Since

L

is not

modular

, by

definition

it contains

elements

x

,

y

and

z

such that

x

≤

z

and

x

∨

(

y

∧

z

)

<

(

x

∨

y

)

∧

z

. Then the sublattice formed by

y

,

x

∨

y

,

y

∧

z

,

(

x

∨

y

)

∧

z

and

x

∨

(

y

∧

z

)

is isomorphic to

N

5

. This is because

y

∧

z

≤

x

∨

(

y

∧

z

)

<

(

x

∨

y

)

∧

z

≤

x

∨

y

while

[

x

∨

(

y

∧

z

)

]

∨

y

=

x

∨

y

by absorption and similarly

y

∧

[

(

x

∨

y

)

∧

z

]

=

y

∧

z

. Moreover,

x

∨

(

y

∧

z

)

covers

y

∧

z

since

y

∧

z

=

x

∨

(

y

∧

z

)

would imply

x

≤

y

∧

z

≤

y

, whence

x

∨

y

=

y

and

(

x

∨

y

)

∧

z

=

y

∧

z

=

x

∨

(

y

∧

z

)

contrary to our

hypothesis

. By the same method,

(

x

∨

y

)

∧

z

=

x

∨

y

leads to a

contradiction

of nonmodularity, so

x

∨

y

covers

(

x

∨

y

)

∧

z

.
∎

Title

nonmodular sublattice

Canonical

name

NonmodularSublattice

Date of creation

2013-03-22 16:55:25

Last modified on

2013-03-22 16:55:25

Owner

ixionid (16766)

Last modified by

ixionid (16766)

Numerical id

8

Author

ixionid (16766)

Entry type

Theorem

Classification

msc 06C05

Related topic

ModularLattice

Generated on Fri Feb 9 09:59:25 2018 by

LaTeXML