aarjaneiro/planetmath huggingface dataset:
Z_2

Z

2

Z

2

is the full system of

second order arithmetic

, that is, the full theory of numbers and sets of numbers. It is

sufficient

for a great deal of mathematics, including much of

number theory

and

analysis

.

The axioms defining

successor

,

addition

,

multiplication

, and comparison are the same as those of PA.

Z

2

adds the full

induction axiom

and the full

comprehension axiom

.

Title

Z

2

Canonical

name

Z2

Date of creation

2013-03-22 12:56:57

Last modified on

2013-03-22 12:56:57

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

5

Author

Henry (455)

Entry type

Definition

Classification

msc 03F35

Synonym

Z

Synonym

Z2

Defines

second order arithmetic

Generated on Thu Feb 8 20:24:47 2018 by

LaTeXML
disjunction property

disjunction property

The

disjunction property

(or

DP

for short) is the meta-statement in logic, which says

if

⊢

A

∨

B

, then

⊢

A

or

⊢

B

.

DP fails for classical

propositional logic

, but is true for intuitionisitc propositional logic. In fact, there are infinitely many intermediate logics between classical and

intuitionistic logics

that

satisfy

DP. Furthermore, there are no intermediate logics maximal with respect to satisfying DP. With respect to predicate logic, DP is ture in

first order

intuitionistic logic without

function symbols

.

There is also a modal version of the disjunction property (or

MDP

for short), which states:

if

⊢

□

⁢

A

∨

□

⁢

B

, then

⊢

A

or

⊢

B

.

It is not hard to see that MDP holds in

normal modal logics

K, T, K4, S4, and GL, and fails in D, B, and S5.

Remark

. In predicate logic, there is also a sort of infinitary analog of DP called the existence property (EP), or the witness property, which states:

if

⊢

∃

x

⁢

P

⁢

(

x

)

, then there is a closed term

t

such that

⊢

P

⁢

(

t

)

.

Like DP, EP fails in classical

first-order logic

, but true in first-order intuitionistic logic without function symbols and with at least one constant symbol.

Title

disjunction property

Canonical name

DisjunctionProperty

Date of creation

2013-03-22 19:35:37

Last modified on

2013-03-22 19:35:37

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

11

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03F55

Classification

msc 03B55

Classification

msc 03B20

Synonym

DP

Related topic

AxiomSystemForIntuitionisticLogic

Related topic

NaturalDeductionForIntuitionisticPropositionalLogic

Related topic

NormalModalLogic

\@unrecurse

Generated on Thu Feb 8 20:25:19 2018 by

LaTeXML
inhabited set

inhabited set

A set

A

is called

inhabited

, if there exists an

element

a

∈

A

. Note that in classical mathematics this is

equivalent

to

A

≠

∅

(i.e.

A

being nonempty), yet in intuitionistic mathematics we actually have to find an element

a

∈

A

.
For example the set, which contains

1

if Goldbach’s

conjecture

is true and

0

if it is false is certainly nonempty, yet by today’s state of knowledge we cannot say if

A

is inhabited, since we do not know an element of

A

.

Title

inhabited set

Canonical

name

InhabitedSet

Date of creation

2013-03-22 14:25:24

Last modified on

2013-03-22 14:25:24

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

6

Author

mathwizard (128)

Entry type

Definition

Classification

msc 03F55

Generated on Thu Feb 8 20:25:20 2018 by

LaTeXML
computable real function

computable real function

A

function

f

:

ℝ

→

ℝ

is

sequentially computable

if, for every

computable sequence

{

x

i

}

i

=

1

∞

of real numbers, the

sequence

{

f

⁢

(

x

i

)

}

i

=

1

∞

is also

computable

.

A function

f

:

ℝ

→

ℝ

is

effectively uniformly continuous

if there exists a

recursive function

d

:

ℕ

→

ℕ

such that, if

|

x

-

y

|

<

1

d

⁢

(

n

)

then

|

f

⁢

(

x

)

-

f

⁢

(

y

)

|

<

1

n

A

real function

is

computable

if it is both sequentially computable and effectively uniformly continuous.

It is not hard to generalize these definitions to functions of more than one variable or functions only defined on a subset of

ℝ

n

. The

generalizations

of the latter two definitions are so obvious that they need not be restated. A suitable generalization of the first definition is:

Let

D

be a subset of

ℝ

n

. A function

f

:

D

→

ℝ

is

sequentially computable

if, for every

n

-tuplet

(

{

x

i

⁢

1

}

i

=

1

∞

,

…

⁢

{

x

i

⁢

n

}

i

=

1

∞

)

of computable sequences of real numbers such that

(

∀

i

)

(

x

i

⁢

1

,

…

⁢

x

i

⁢

n

)

∈

D

,

the sequence

{

f

⁢

(

x

i

)

}

i

=

1

∞

is also computable.

Title

computable real function

Canonical name

ComputableRealFunction

Date of creation

2013-03-22 14:39:23

Last modified on

2013-03-22 14:39:23

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

6

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 03F60

Defines

sequentially computable

Defines

effectively uniformly continuous

Defines

effective uniform continuity

Generated on Thu Feb 8 20:25:23 2018 by

LaTeXML
computable sequence

computable sequence

A

sequence

{

r

i

}

i

=

0

∞

of

rational numbers

is called

computable

if there exist

recursive functions

F

:

ℕ

→

ℤ

and

G

:

ℕ

→

ℕ

∖

{

0

}

such that

r

i

=

F

⁢

(

i

)

G

⁢

(

i

)

.

A sequence

{

r

i

}

i

=

0

∞

of real numbers is called

computable

if there exists a recursive functions

F

:

ℕ

2

→

ℤ

such that

(

∀

m

>

0

)

(

∀

n

)

F

⁢

(

m

,

n

)

-

1

n

<

r

m

<

F

⁢

(

m

,

n

)

+

1

n

.

Note that, although every

computable sequence

of real numbers is a sequence of computable real numbers, not every sequence of computable real numbers is a computable sequence of real numbers. Formally, this is the case because the set of computable sequences of real numbers is

countable

(its

cardinality

cannot be greater than that of the set of recursive functions) whilst the set of sequences of computable real numbers is

uncountable

(by Cantor’s diagonal argument).

An explanation of this fact which, if somewhat less

rigorous

, is more satisfying to the intuition is the following: Imagine a sequence of computable real numbers such that the shortest FORTRAN program which could calculate the first number to arbitrarily specified precision is at least 1 kilobyte long, the shortest FORTRAN program which which could calculate the second number to arbitrarily specified precision is at least 2 kilobytes long, the shortest FORTRAN program which could calculate the third number to arbitrarily specified precision is at least 3 kilobytes long, and so on. If this sequence were computable, there would exist a finite FORTRAN program that would compute the recursive function

F

that appears in the definition of computable sequence. By adding a few more lines of to such a program, one would have a program that could compute any element of the sequence to arbitrary precision. However, no such program could possibly exist because of the way that the sequence was defined, and hence this sequence is not a computable sequence even though each of its

elements

is a computable real number.

Title

computable sequence

Canonical name

ComputableSequence

Date of creation

2013-03-22 14:39:20

Last modified on

2013-03-22 14:39:20

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

13

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 03F60

Generated on Thu Feb 8 20:25:25 2018 by

LaTeXML
algebraic definition of a lattice

algebraic definition of a lattice

The parent entry (

http://planetmath.org/

Lattice

) defines a lattice as a

relational structure

(a poset) satisfying the condition that every pair of

elements

has a

supremum

and an

infimum

. Alternatively and equivalently, a lattice

L

can be a defined directly as an

algebraic structure

with two

binary operations

called meet

∧

and join

∨

satisfying the following conditions:

•

(

idempotency

of

∨

and

∧

): for each

a

∈

L

,

a

∨

a

=

a

∧

a

=

a

;

•

(

commutativity

of

∨

and

∧

): for every

a

,

b

∈

L

,

a

∨

b

=

b

∨

a

and

a

∧

b

=

b

∧

a

;

•

(

associativity

of

∨

and

∧

): for every

a

,

b

,

c

∈

L

,

a

∨

(

b

∨

c

)

=

(

a

∨

b

)

∨

c

and

a

∧

(

b

∧

c

)

=

(

a

∧

b

)

∧

c

; and

•

(absorption): for every

a

,

b

∈

L

,

a

∧

(

a

∨

b

)

=

a

and

a

∨

(

a

∧

b

)

=

a

.

It is easy to see that this

definition

is

equivalent

to the one given in the parent, as follows: define a

binary relation

≤

on

L

such that

a

≤

b

iff

a

∨

b

=

b

.

Then

≤

is

reflexive

by the idempotency of

∨

. Next, if

a

≤

b

and

b

≤

a

, then

a

=

a

∨

b

=

b

, so

≤

is anti-symmetric. Finally, if

a

≤

b

and

b

≤

c

, then

a

∨

c

=

a

∨

(

b

∨

c

)

=

(

a

∨

b

)

∨

c

=

b

∨

c

=

c

, and therefore

a

≤

c

. So

≤

is

transitive

. This shows that

≤

is a

partial order

on

L

. For any

a

,

b

∈

L

,

a

∨

(

a

∨

b

)

=

(

a

∨

a

)

∨

b

=

a

∨

b

so that

a

≤

a

∨

b

. Similarly,

b

≤

a

∨

b

. If

a

≤

c

and

b

≤

c

, then

(

a

∨

b

)

∨

c

=

a

∨

(

b

∨

c

)

=

a

∨

c

=

c

. This shows that

a

∨

b

is the supremum of

a

and

b

. Similarly,

a

∧

b

is the infimum of

a

and

b

.

Conversely

, if

(

L

,

≤

)

is defined as in the parent entry, then by defining

a

∨

b

=

sup

⁡

{

a

,

b

}

and

a

∧

b

=

inf

⁡

{

a

,

b

}

,

the four conditions above are satisfied. For example, let us show one of the

absorption laws

:

a

∨

(

a

∧

b

)

=

a

. Let

c

=

inf

⁡

{

a

,

b

}

≤

a

=

a

∧

b

. Then

c

≤

a

so that

sup

⁡

{

a

,

c

}

=

a

, which precisely translates to

a

=

a

∨

c

=

a

∨

(

a

∧

b

)

. The

remainder

of the proof is left for the reader to try.

Title

algebraic definition of a lattice

Canonical

name

AlgebraicDefinitionOfALattice

Date of creation

2013-03-22 17:39:29

Last modified on

2013-03-22 17:39:29

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

12

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03G10

Classification

msc 06B99

Generated on Thu Feb 8 20:26:07 2018 by

LaTeXML
complete Heyting algebra

complete Heyting algebra

A

Heyting algebra

that is also a

complete lattice

is called a

complete Heyting algebra

. In the following, we give a

lattice

characterization

of complete Heyting algebras without the relative pseudocomplementation

operator

→

.

Proposition 1

.

Let

H

be a complete Heyting algebra, then

x

→

y

=

⋁

{

z

∣

z

∧

x

≤

y

}

for any

x

,

y

∈

H

, and

x

∧

⋁

A

=

⋁

(

x

∧

A

)

,

where

x

∧

A

:=

{

x

∧

y

∣

y

∈

A

}

, for any

x

∈

H

, and any subset

A

of

H

.

Proof.

We first prove the

identity

x

∧

⋁

A

=

⋁

(

x

∧

A

)

. For any

y

∈

H

, we have

x

∧

⋁

A

≤

y

iff

⋁

A

≤

x

→

y

iff

a

≤

x

→

y

for all

a

∈

A

iff

x

∧

a

≤

y

for all

a

∈

A

iff

⋁

(

x

∧

A

)

≤

y

, hence

x

∧

⋁

A

=

⋁

(

x

∧

A

)

.

Next, we show

x

→

y

=

⋁

{

z

∣

z

∧

x

≤

y

}

. For any

a

∈

H

, we have

a

≤

x

→

y

iff

a

∧

x

≤

y

iff

a

∈

{

z

∣

z

∧

x

≤

y

}

iff

a

≤

⋁

{

z

∣

z

∧

x

≤

y

}

.
∎

The

converse

of the above is also true.

Proposition 2

.

Let

H

be a complete lattice such that

x

∧

⋁

A

=

⋁

(

x

∧

A

)

,

where

x

∧

A

:=

{

x

∧

y

∣

y

∈

A

}

, for any

x

∈

H

, and any subset

A

of

H

. Then for any

x

,

y

∈

H

, defining

x

→

y

:=

⋁

{

z

∣

z

∧

x

≤

y

}

turns

H

into a complete Heyting algebra.

Proof.

We want to show that

a

≤

x

→

y

iff

a

∧

x

≤

y

for any

a

∈

H

:

a

≤

x

→

y

iff

a

≤

⋁

{

z

∣

z

∧

x

≤

y

}

. So

x

∧

a

≤

x

∧

⋁

{

z

∣

z

∧

x

≤

y

}

=

⋁

{

x

∧

z

∣

z

∧

x

≤

y

}

≤

⋁

{

y

}

=

y

∎

From this, one readily concludes that any finite

distributive lattice

is Heyting.

Remark

. Since any complete lattice is

bounded

, a

complete

Brouwerian lattice

is a complete Heyting algebra. A complete Heyting algebra is also called a

frame

.

Title

complete Heyting algebra

Canonical

name

CompleteHeytingAlgebra

Date of creation

2013-03-22 19:31:42

Last modified on

2013-03-22 19:31:42

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03G10

Classification

msc 06D20

Related topic

Locale

Generated on Thu Feb 8 20:26:19 2018 by

LaTeXML
consequence operator is determined by its fixed points

consequence operator is determined by its fixed points

Theorem 1

Suppose that

C

1

and

C

2

are

consequence operators

on a set

L

and that,
for every

X

⊆

L

, it happens that

C

1

⁢

(

X

)

=

X

if and only if

C

2

⁢

(

X

)

=

X

. Then

C

1

=

C

2

.

Theorem 2

Suppose that

C

is a consequence operators on a set

L

. Define

K

=

{

X

⊆

L

∣

C

⁢

(

X

)

=

X

}

. Then, for every

X

∈

L

, there exists a

Y

∈

K

such that

X

⊆

Y

and, for every

Z

∈

K

such that

X

⊆

Z

,
one has

Y

⊆

Z

.

Theorem 3

Given a set

L

, suppose that

K

is a subset of

L

such that, for every

X

∈

L

, there exists a

Y

∈

K

such that

X

⊆

Y

and, for every

Z

∈

K

such that

X

⊆

Z

, one has

Y

⊆

Z

. Then there exists a
consequence operator

C

:

P

⁢

(

L

)

→

P

⁢

(

L

)

such that

C

⁢

(

X

)

=

X

if and only if

X

∈

K

.

Title

consequence operator is determined by its fixed points

Canonical name

ConsequenceOperatorIsDeterminedByItsFixedPoints

Date of creation

2013-03-22 16:29:48

Last modified on

2013-03-22 16:29:48

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

6

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 03G10

Classification

msc 03B22

Classification

msc 03G25

Generated on Thu Feb 8 20:26:21 2018 by

LaTeXML
operations on consequence operators

operations on consequence operators

Let

L

be a set and let

𝒞

be the set of all

consequence operators

on

S

. Then we may define a

binary relation

≤

⊂

𝒞

×

𝒞

and

binary operations

∧

,

∨

,

⊻

:

𝒞

×

𝒞

→

𝒞

as follows:

Definition 1

For

C

1

,

C

2

∈

C

, we have

C

1

≤

C

2

when, for all

X

⊆

L

, we have

C

1

⁢

(

X

)

⊆

C

2

⁢

(

X

)

Definition 2

For

C

1

,

C

2

∈

C

, we have

(

C

1

∧

C

2

)

⁢

(

X

)

=

C

1

⁢

(

X

)

∩

C

2

⁢

(

X

)

for all

X

⊆

L

.

Definition 3

For

C

1

,

C

2

∈

C

, we have

(

C

1

∨

C

2

)

⁢

(

X

)

=

C

1

⁢

(

X

)

∪

C

2

⁢

(

X

)

for all

X

⊆

L

.

Definition 4

For

C

1

,

C

2

∈

C

, we have

(

C

1

⊻

C

2

)

⁢

(

X

)

=

∩

{

Y

∣

X

⊆

Y

⊆

L

∧

C

1

⁢

(

Y

)

=

C

2

⁢

(

Y

)

=

Y

}

for all

X

⊆

L

.

Title

operations on consequence operators

Canonical name

OperationsOnConsequenceOperators

Date of creation

2013-03-22 16:29:38

Last modified on

2013-03-22 16:29:38

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

6

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 03G10

Classification

msc 03B22

Classification

msc 03G25

Generated on Thu Feb 8 20:26:37 2018 by

LaTeXML
Jan Łukasiewicz

Jan Łukasiewicz

Jan Łukasiewicz

(1878 - 1956) Polish mathematician and logician mainly concerned with logic in mathematics probably best known for the

three-valued logics

, the

Polish notation

, the

law of excluded middle

and the axiomatizations of much classical

propositional logic

. He studied at the University of Lwów, earning doctorates in mathematics and philosophy in 1902. In 1919 he was Minister of Education in Poland. After WWII, he was exiled to Belgium. Arguably, the 3-valued logics named after him was the first published report of a non-Boolean, or non-Chrysippean logic (with only two logic values, ‘true’ or ‘false’, in the Chrysippean case). Subsequently, Łukasiewicz

logic algebras

were constructed by Grigore Moisil in 1940-1945 to define ‘nuances’ in logics, or

many-valued logics

, as well as 3-state control logic (electronic)

circuits

. Łukasiewicz-Moisil (

L

⁢

M

n

) logic algebras were defined axiomatically after 1969 as n-valued logic algebra representations and

extensions

of the Łukasiewcz (3-valued) logics; then, the

universal properties

of

categories

of

L

⁢

M

n

-logic algebras (

http://planetmath.org/AlgebraicCategoryOfLMnLogicAlgebras

) were also investigated and reported in a series of recent publications. Recently, several modifications of

L

⁢

M

n

-logic algebras

are under consideration as valid candidates for

representations

of

quantum logics

, as well as for modeling non-linear biodynamics in genetic ‘nets’ or

networks

, as well as in single-cell organisms, or in tumor growth.

Title

Jan Łukasiewicz

Canonical name

JanLukasiewicz

Date of creation

2013-03-22 16:10:12

Last modified on

2013-03-22 16:10:12

Owner

Mravinci (12996)

Last modified by

Mravinci (12996)

Numerical id

17

Author

Mravinci (12996)

Entry type

Definition

Classification

msc 03G12

Classification

msc 03G30

Classification

msc 03G10

Classification

msc 92B05

Classification

msc 01A60

Classification

msc 03G20

Synonym

Jan Ł ukasiewicz

Synonym

Jan Łukasiewicz

Synonym

Jan Lukasiewicz

Related topic

GeneticNetsOrNetworks

Related topic

AlgebraicCategoryOfLMnLogicAlgebras

Related topic

AnalyticsAndOntologyFormalLogics

Related topic

FormalLogicsAndMetaMathematics

Generated on Thu Feb 8 20:26:51 2018 by

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example of polyadic algebra with equality

example of polyadic algebra with equality

Recall that given a triple

(

A

,

I

,

X

)

where

A

is a

Boolean algebra

,

I

and

X

≠

∅

are sets. we can construct a

polyadic algebra

(

B

,

I

,

∃

,

S

)

called the

functional polyadic algebra

for

(

A

,

I

,

X

)

. In this entry, we will construct an example of a

polyadic algebra with equality

called the

functional polyadic algebra with equality

from

(

B

,

I

,

∃

,

S

)

.

We start with a simpler

structure

. Let

B

be an arbitrary Boolean algebra,

I

and

X

≠

∅

are sets. Let

Y

=

X

I

, the set of all

I

-indexed

X

-valued

sequences

, and

Z

=

B

Y

, the set of all

functions

from

Y

to

B

. Call the function

e

:

I

×

I

→

Z

the

functional equality associated with

(

B

,

I

,

X

)

, if for each

i

,

j

∈

I

,

e

⁢

(

i

,

j

)

is the function defined by

e

⁢

(

i

,

j

)

⁢

(

x

)

:=

{

1

if

⁢

x

i

=

x

j

,

0

otherwise.

The quadruple

(

B

,

I

,

X

,

e

)

is called a

functional equality algebra

.

Now,

B

will have the additional structure of being a polyadic algebra. Start with a Boolean algebra

A

, and let

I

and

X

be defined as in the last paragraph. Then, as stated above in the first paragraph, and illustrated in here (

http://planetmath.org/ExampleOfPolyadicAlgebra

),

(

B

,

I

,

∃

,

S

)

is a polyadic algebra (called the functional polyadic algebra for

(

A

,

I

,

X

)

). Using the

B

just constructed, the quadruple

(

B

,

I

,

X

,

e

)

is a functional equality algebra, and is called the

functional polyadic algebra with equality

for

(

A

,

I

,

X

)

.

It is not hard to show that

e

is an equality predicate on

C

=

(

B

,

I

,

∃

,

S

)

, and as a result

(

C

,

e

)

is a polyadic algebra with equality.

References

1

P. Halmos,

Algebraic Logic

, Chelsea Publishing Co. New York (1962).

Title

example of polyadic algebra with equality

Canonical

name

ExampleOfPolyadicAlgebraWithEquality

Date of creation

2013-03-22 17:55:24

Last modified on

2013-03-22 17:55:24

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Example

Classification

msc 03G15

Defines

functional equality algebra

Defines

functional equality

Defines

functional polyadic algebra with equality

Generated on Thu Feb 8 20:28:03 2018 by

LaTeXML
special elements in a relation algebra

special elements in a relation algebra

Let

A

be a

relation algebra

with

operators

(

∨

,

∧

,

;

,

′

,

-

,

0

,

1

,

i

)

of type

(

2

,

2

,

2

,

1

,

1

,

0

,

0

,

0

)

. Then

a

∈

A

is called a

•

function element

if

e

-

;

e

≤

i

,

•

injective element

if it is a function element such that

e

;

e

-

≤

i

,

•

surjective element

if

e

-

;

e

=

i

,

•

reflexive element

if

i

≤

a

,

•

symmetric element

if

a

-

≤

a

,

•

transitive element

if

a

;

a

≤

a

,

•

subidentity

if

a

≤

i

,

•

antisymmetric element

if

a

∧

a

-

is a subidentity,

•

equivalence element

if it is

symmetric

and

transitive

(not necessarily

reflexive

!),

•

domain element

if

a

;

1

=

a

,

•

range element

if

1

;

a

=

a

,

•

ideal element

if

1

;

a

;

1

=

a

,

•

rectangle

if

a

=

b

;

1

;

c

for some

b

,

c

∈

A

, and

•

square

if it is a rectangle where

b

=

c

(using the notations above).

These special

elements

are so named because they are the names of the corresponding

binary relations

on a set. The following table shows the correspondence.

element in relation algebra

A

binary relation on set

S

function element

function

(on

S

)

injective element

injection

surjective element

surjection

reflexive element

reflexive relation

symmetric element

symmetric relation

transitive element

transitive relation

subidentity

I

T

:=

{

(

x

,

x

)

∣

x

∈

T

}

where

T

⊆

S

antisymmetric element

antisymmetric relation

equivalence element

symmetric reflexive relation (not an

equivalence relation

!)

domain element

dom

⁡

(

R

)

×

S

where

R

⊆

S

2

range element

S

×

ran

⁡

(

R

)

where

R

⊆

S

2

ideal element

rectangle

U

×

V

⊆

S

2

square

U

2

, where

U

⊆

S

References

1

⁢

S

.

R

.

G

⁢

i

⁢

v

⁢

a

⁢

n

⁢

t

,

The Structure of Relation Algebras Generated by Relativizations

,

A

⁢

m

⁢

e

⁢

r

⁢

i

⁢

c

⁢

a

⁢

n

⁢

M

⁢

a

⁢

t

⁢

h

⁢

e

⁢

m

⁢

a

⁢

t

⁢

i

⁢

c

⁢

a

⁢

l

⁢

S

⁢

o

⁢

c

⁢

i

⁢

e

⁢

t

⁢

y

⁢

(

1994

)

.

Title

special elements in a relation algebra

Canonical name

SpecialElementsInARelationAlgebra

Date of creation

2013-03-22 17:48:43

Last modified on

2013-03-22 17:48:43

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03G15

Defines

function element

Defines

injective element

Defines

surjective element

Defines

reflexive element

Defines

symmetric element

Defines

transitive element

Defines

equivalence element

Defines

domain element

Defines

range element

Defines

ideal element

Defines

rectangle

Defines

square

Defines

antisymmetric element

Defines

subidentity

Generated on Thu Feb 8 20:29:08 2018 by

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