aarjaneiro/planetmath huggingface dataset:
graded poset

graded poset

A

graded poset

is a poset

P

that is equipped with a

rank function

ρ

, which is a

function

from

P

to

ℤ

, satisfying the following three conditions:

1.

ρ

is

constant

on all

minimal elements

of

P

, usually with value

-

1

or

0

2.

ρ

is

isotone

, that is, if

a

≤

b

, then

ρ

⁢

(

a

)

≤

ρ

⁢

(

b

)

, and

3.

ρ

preserves

covering relations: if

a

≺

b

, then

ρ

⁢

(

a

)

+

1

=

ρ

⁢

(

b

)

.

Equivalently, a poset

P

is graded if it admits a

partition

into

maximal antichains

{

A

n

∣

n

∈

ℕ

}

such that
for each

x

∈

A

n

,
all of the

elements

covering

x

are in

A

n

+

1

and all the elements
covered by

x

are in

A

n

-

1

.

A poset

P

can be graded

if one can define a rank function

ρ

on

P

so

(

P

,

ρ

)

is a graded poset. Below is a poset that can not be graded:

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Generalized rank functions

Since certain common posets such as the face

lattice

of a polytope are most naturally graded by

dimension

(

http://planetmath.org/Dimension2

), the rank of a minimal element is sometimes required to be

-

1

.

More generally, given a chain

C

, one can define

C

-graded posets

. A poset

P

is

C

-graded provided that there is a poset map

ρ

:

P

→

C

that preserves covers and is constant on minimal elements of

P

. Such a rank function is unique up to choice of the rank of minimal elements. In practice, however, the term graded is only used to indicate

ℕ

-grading,

ℕ

∪

{

-

1

}

-grading, or

ℤ

-grading.

Maximal chains in graded posets

Let

P

be a graded poset with rank function

ρ

. A chain

C

in

P

is said to be a

saturated chain

provided that

ρ

⁢

(

C

)

=

ρ

⁢

(

P

)

. If

C

is

saturated

in

P

, then each

cover relation

in

C

is also a cover relation in

P

; thus a saturated chain is also a maximal chain.

It is a

property

of graded posets that all saturated chains have the same

cardinality

. As a partial

converse

, if

P

is a finite

bounded poset

(

http://planetmath.org/BoundedLattice

) and each maximal chain has the same cardinality, then

P

is graded.

Title

graded poset

Canonical

name

GradedPoset

Date of creation

2013-03-22 14:09:12

Last modified on

2013-03-22 14:09:12

Owner

mps (409)

Last modified by

mps (409)

Numerical id

9

Author

mps (409)

Entry type

Definition

Classification

msc 06A06

Classification

msc 05B35

Related topic

EulerianPoset

Related topic

StarProduct

Related topic

HeightOfAnElementInAPoset

Defines

rank function

Defines

saturated chain

Generated on Fri Feb 9 09:51:55 2018 by

LaTeXML
height of an element in a poset

height of an element in a poset

Let

P

be a poset. Given any

a

∈

P

, the

lower set

↓

a

of

a

is a subposet of

P

. Call the height of

↓

a

less 1 the

height

of

a

. Let’s denote

h

⁢

(

a

)

the height of

a

, so

h

(

a

)

=

height

(

↓

a

)

-

1

.

From this

definition

, we see that

h

⁢

(

a

)

=

0

iff

a

is

minimal

and

h

⁢

(

a

)

=

1

iff

a

is an atom. Also,

h

partitions

P

into

equivalence classes

, so that

a

is

equivalent

to

b

in

P

iff

h

⁢

(

a

)

=

h

⁢

(

b

)

. Two distinct

elements

in the same equivalence class are necessarily

incomparable

. In other words, the equivalence classes are

antichains

. Furthermore, given any two equivalence classes

[

a

]

,

[

b

]

, set

[

a

]

≤

[

b

]

iff

h

⁢

(

a

)

≤

h

⁢

(

b

)

, then the set of equivalence classes form a chain.

The height function of a poset

P

looks remarkably like the

rank function

of a graded poset:

h

is

constant

on the set of all

minimal elements

, and

h

is

isotone

(

preserves

order). When is

h

a rank function (the additional condition being the preservation of the covering relation)? The answer is given by a

chain condition

imposed on

P

, called the

Jordan-Dedekind chain condition

:

(*) In a poset, the

cardinalities

of two maximal chains between common

end points

must be the same.

Suppose for each

a

∈

P

,

h

⁢

(

a

)

is finite and

P

has a unique minimal element

0

. Then

P

can be graded by

h

iff (*) is satisfied. More generally, if we drop the

assumption

of the uniqueness of a minimal element, then

P

can be graded by

h

iff any two maximal chains ending at the same end point have the same length.

Title

height of an element in a poset

Canonical

name

HeightOfAnElementInAPoset

Date of creation

2013-03-22 16:31:32

Last modified on

2013-03-22 16:31:32

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06A06

Related topic

GradedPoset

Defines

Jordan-Dedekind chain condition

Generated on Fri Feb 9 09:52:06 2018 by

LaTeXML
infimum

infimum

The

infimum

of a set

S

is the

greatest lower bound

of

S

and is denoted

inf

⁡

(

S

)

.

Let

A

be a set with a

partial order

≤

, and let

S

⊆

A

. For any

x

∈

A

,

x

is a

lower bound

of

S

if

x

≤

y

for any

y

∈

S

. The infimum of

S

, denoted

inf

⁡

(

S

)

, is the greatest such lower bound; that is, if

b

is a lower bound of

S

, then

b

≤

inf

⁡

(

S

)

.

Note that it is not necessarily the case that

inf

⁡

(

S

)

∈

S

. Suppose

S

=

(

0

,

1

)

; then

inf

⁡

(

S

)

=

0

, but

0

∉

S

.

Also note that a set does not necessarily have an infimum. See the attachments to this entry for examples.

Title

infimum

Canonical

name

Infimum

Date of creation

2013-03-22 11:48:09

Last modified on

2013-03-22 11:48:09

Owner

vampyr (22)

Last modified by

vampyr (22)

Numerical id

11

Author

vampyr (22)

Entry type

Definition

Classification

msc 06A06

Classification

msc 03D20

Related topic

Supremum

Related topic

LebesgueOuterMeasure

Related topic

MinimalAndMaximalNumber

Related topic

InfimumAndSupremumForRealNumbers

Related topic

NondecreasingSequenceWithUpperBound

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

LaTeXML
order-preserving map

order-preserving map

Order-preserving map

from a poset

L

to a poset

M

is a

function

f

such that

∀

x

,

y

∈

L

:

(

x

≥

y

⟹

f

(

x

)

≥

f

(

y

)

)

.

Order-preserving maps are also called

monotone functions

or

monotonic functions

or

order homomorphisms

or

isotone functions

or

isotonic functions

.

Order-reversing map

from a poset

L

to a poset

M

is a function

f

such that

∀

x

,

y

∈

L

:

(

x

≥

y

⟹

f

(

x

)

≤

f

(

y

)

)

.

Order-reversing maps are also called

antitone functions

.

Title

order-preserving map

Canonical

name

OrderpreservingMap

Date of creation

2013-03-22 17:44:43

Last modified on

2013-03-22 17:44:43

Owner

porton (9363)

Last modified by

porton (9363)

Numerical id

10

Author

porton (9363)

Entry type

Definition

Classification

msc 06A06

Synonym

monotone function

Synonym

monotonic function

Synonym

order homomorphism

Synonym

isotone function

Synonym

isotonic function

Synonym

order-preserving

Synonym

isotone

Synonym

isotonic

Synonym

order-reversing

Synonym

antitonic

Synonym

antitone

Related topic

Poset

Related topic

LatticeHomomorphism

Defines

monotonicity

Generated on Fri Feb 9 09:52:27 2018 by

LaTeXML
partial order

partial order

A

partial order

(often simply referred to as an

order

or

ordering

) is a

relation

≤

⊂

A

×

A

that

satisfies

the following three properties:

1.

Reflexivity

:

a

≤

a

for all

a

∈

A

2.

Antisymmetry: If

a

≤

b

and

b

≤

a

for any

a

,

b

∈

A

, then

a

=

b

3.

Transitivity

: If

a

≤

b

and

b

≤

c

for any

a

,

b

,

c

∈

A

, then

a

≤

c

A

total order

is a partial order that satisfies a fourth property known as

comparability

:

•

Comparability: For any

a

,

b

∈

A

, either

a

≤

b

or

b

≤

a

.

A set and a partial order on that set define a poset.

Remark

. In some literature, especially those dealing with the

foundations of mathematics

, a partial order

≤

is defined as a

transitive

irreflexive

binary relation

(on a set). As a result, if

a

≤

b

, then

b

≰

a

, and therefore

≤

is antisymmetric.

Title

partial order

Canonical name

PartialOrder

Date of creation

2013-03-22 11:43:32

Last modified on

2013-03-22 11:43:32

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

24

Author

mathcam (2727)

Entry type

Definition

Classification

msc 06A06

Classification

msc 35C10

Classification

msc 35C15

Classification

msc 55-01

Classification

msc 55-00

Synonym

order

Synonym

partial ordering

Synonym

ordering

Related topic

Relation

Related topic

TotalOrder

Related topic

Poset

Related topic

BinarySearch

Related topic

SortingProblem

Related topic

ChainCondition

Related topic

PartialOrderWithChainConditionDoesNotCollapseCardinals

Related topic

QuasiOrder

Related topic

CategoryAssociatedToAPartialOrder

Related topic

OrderingRelation

Related topic

HasseDiagram

Related topic

NetsAndClosuresOfSubspaces

Generated on Fri Feb 9 09:52:29 2018 by

LaTeXML
poset height and width

poset height and width

Let

P

be a poset. Then the

width

of

P

is the

cardinality

of a largest (maximum cardinality)

antichain

in

P

and the

height

of

P

is the cardinality of a longest (maximum cardinality) chain in

P

.

Title

poset height and width

Canonical name

PosetHeightAndWidth

Date of creation

2013-03-22 15:01:46

Last modified on

2013-03-22 15:01:46

Owner

justice (4961)

Last modified by

justice (4961)

Numerical id

7

Author

justice (4961)

Entry type

Definition

Classification

msc 06A06

Defines

height

Defines

width

Generated on Fri Feb 9 09:52:31 2018 by

LaTeXML
pure poset

pure poset

A poset is

pure

if it is finite and every maximal chain has the same length.
If

P

is a

pure poset

, we can create a

rank function

r

on

P

by defining

r

(

x

) to be the length of a maximal chain bounded above by

x

. Every
interval of a pure poset is a graded poset, and every graded poset is pure. Moreover,
the

closure

of a pure poset, formed by adjoining an artificial minimum element and
an artificial maximum element, is always graded.

The face poset of a

pure simplicial complex

is pure as a poset.

Title

pure poset

Canonical name

PurePoset

Date of creation

2013-03-22 17:19:47

Last modified on

2013-03-22 17:19:47

Owner

mps (409)

Last modified by

mps (409)

Numerical id

4

Author

mps (409)

Entry type

Definition

Classification

msc 06A06

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

LaTeXML
sets that do not have an infimum

sets that do not have an infimum

Some examples for sets that do not have an

infimum

:

•

The set

M

1

:=

ℚ

(as a subset of

ℚ

) does not have
an infimum (nor a

supremum

). Intuitively this is clear, as the set is

unbounded

. The (easy) formal proof is left as an exercise for the reader.

•

A more interesting example: The set

M

2

:=

{

x

∈

ℚ

:

x

2

≥

2

,

x

>

0

}

(again as a subset of

ℚ

) .

Proof.

Clearly,

inf

⁡

(

M

2

)

>

0

. Assume

i

>

0

is an infimum of

M

2

. Now we use
the fact that

2

is not

rational

, and therefore

i

<

2

or

i

>

2

.

If

i

<

2

, choose any

j

∈

ℚ

from the

interval

(

i

,

2

)

⊂

ℝ

(this is a real interval, but as the

rational numbers

are dense (

http://

planetmath

.org/Dense

) in the

real numbers

, every nonempty interval in

ℝ

contains a rational number, hence such a

j

exists).

Then

j

>

i

, but

j

<

2

, hence

j

2

<

2

and therefore

j

is a

lower
bound

for

M

2

, which is a

contradiction

.

On the other hand, if

i

>

2

, the

argument

is very

similar

:
Choose any

j

∈

ℚ

from the interval

(

2

,

i

)

⊂

ℝ

. Then

j

<

i

, but

j

>

2

, hence

j

2

>

2

and therefore

j

∈

M

2

. Thus

M

2

contains an

element

j

smaller than

i

, which is
a contradiction to the

assumption

that

i

=

inf

⁡

(

M

2

)

∎

Intuitively speaking, this example exploits the fact that

ℚ

does
not have “enough elements”. More formally,

ℚ

as a

metric space

is not

complete

(

http://planetmath.org/Complete

). The

M

2

defined above is the real interval

M

2

′

:=

(

2

,

∞

)

⊂

ℝ

intersected with

ℚ

.

M

2

′

as a subset of

ℝ

does have an infimum (namely

2

),
but as that is not an element of

ℚ

,

M

2

does not have an
infimum as a subset of

ℚ

.

This example also makes it clear that it is important to clearly state the

superset

one is working in when using the notion of infimum or supremum.

It also illustrates that the infimum is a natural

generalization

of the

minimum

of a set, as a set that does not have a minimum may still have
an infimum (such as

M

2

′

).

Of course all the ideas expressed here equally apply to the supremum, as the
two notions are completely analogous (just reverse all

inequalities

).

Title

sets that do not have an infimum

Canonical name

SetsThatDoNotHaveAnInfimum

Date of creation

2013-03-22 13:09:51

Last modified on

2013-03-22 13:09:51

Owner

sleske (997)

Last modified by

sleske (997)

Numerical id

9

Author

sleske (997)

Entry type

Example

Classification

msc 06A06

Related topic

InfimumAndSupremumForRealNumbers

Generated on Fri Feb 9 09:53:00 2018 by

LaTeXML
supremum

supremum

The

supremum

of a set

X

having a

partial order

is the least

upper bound

of

X

(if it exists) and is denoted

sup

⁡

X

.

Let

A

be a set with a partial order

⩽

, and let

X

⊆

A

. Then

s

=

sup

⁡

X

if and only if:

1.

For all

x

∈

X

, we have

x

⩽

s

(i.e.

s

is an upper bound).

2.

If

s

′

meets condition

1

, then

s

⩽

s

′

(

s

is the

least

upper bound).

There is another useful

definition

which works if

A

=

ℝ

with

⩽

the usual order on

ℝ

, supposing that s is an upper bound:

s

=

sup

X

if and only if

∀

ε

>

0

,

∃

x

∈

X

:

s

-

ε

<

x

.

Note that it is not necessarily the case that

sup

⁡

X

∈

X

. Suppose

X

=

]

0

,

1

[

, then

sup

⁡

X

=

1

, but

1

∉

X

.

Note also that a set may not have an upper bound at all.

Title

supremum

Canonical

name

Supremum

Date of creation

2013-03-22 11:48:12

Last modified on

2013-03-22 11:48:12

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

11

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06A06

Related topic

Infimum

Related topic

MinimalAndMaximalNumber

Related topic

InfimumAndSupremumForRealNumbers

Related topic

ExistenceOfSquareRootsOfNonNegativeRealNumbers

Related topic

LinearContinuum

Related topic

NondecreasingSequenceWithUpperBound

Related topic

EssentialSupremum

Generated on Fri Feb 9 09:53:02 2018 by

LaTeXML
upper bound

upper bound

Let

S

be a set with a

partial ordering

≤

, and let

T

be a subset of

S

.
An

upper bound

for

T

is an element

z

∈

S

such that

x

≤

z

for all

x

∈

T

. We say that

T

is

bounded from above

if there exists an upper bound for

T

.

Lower bound

, and

bounded from below

are defined in a similar manner.

Title

upper bound

Canonical name

UpperBound

Date of creation

2013-03-22 11:52:15

Last modified on

2013-03-22 11:52:15

Owner

djao (24)

Last modified by

djao (24)

Numerical id

9

Author

djao (24)

Entry type

Definition

Classification

msc 06A06

Classification

msc 11A07

Defines

bound

Defines

lower bound

Defines

bounded

Defines

bounded from above

Defines

bounded from below

Generated on Fri Feb 9 09:53:04 2018 by

LaTeXML
upper set

upper set

Let

P

be a poset and

A

a subset of

P

. The

upper set

of

A

is defined to be the set

{

b

∈

P

∣

a

≤

b

⁢

for some

⁢

a

∈

A

}

,

and is denoted by

↑

A

. In other words,

↑

A

is the set of all

upper bounds

of elements of

A

.

↑

can be viewed as a unary

operator

on the

power set

2

P

sending

A

∈

2

P

to

↑

A

∈

2

P

.

↑

has the following properties

1.

↑

∅

=

∅

,

2.

A

⊆

↑

A

,

3.

↑

↑

A

=

↑

A

, and

4.

if

A

⊆

B

,

↑

A

⊆

↑

B

.

So

↑

is a

closure operator

.

An

upper set

in

P

is a subset

A

such that its upper set is itself:

↑

A

=

A

. In other words,

A

is closed with respect to

≤

in the sense that if

a

∈

A

and

a

≤

b

, then

b

∈

A

. An upper set is also said to be

upper closed

. For this reason, for any subset

A

of

P

, the

↑

A

is also called the

upper closure

of

A

.

Dually, the

lower set

(or

lower closure

) of

A

is the set of all

lower bounds

of elements of

A

. The lower set of

A

is denoted by

↓

A

. If the lower set of

A

is

A

itself, then

A

is a called a

lower set

, or a

lower closed set

.

Remarks

.

•

↑

A

is

not

the same as the set of upper bounds of

A

, commonly denoted by

A

u

, which is defined as the set

{

b

∈

P

∣

a

≤

b

⁢

for

all

⁢

a

∈

A

}

. Similarly,

↓

A

≠

A

ℓ

in general, where

A

ℓ

is the set of lower bounds of

A

.

•

When

A

=

{

x

}

, we write

↑

x

for

↑

A

and

↓

x

for

↓

A

.

↑

x

=

{

x

}

u

and

↓

x

=

{

x

}

d

.

•

If

P

is a

lattice

and

x

∈

P

, then

↑

x

is the

principal filter

generated by

x

, and

↓

x

is the

principal ideal

generated by

x

.

•

If

A

is a lower set of

P

, then its set

complement

A

∁

is an upper set: if

a

∈

A

∁

and

a

≤

b

, then

b

∈

A

∁

by a

contrapositive

argument

.

•

Let

P

be a poset. The set of all lower sets of

P

is denoted by

𝒪

⁢

(

P

)

. It is easy to see that

𝒪

⁢

(

P

)

is a poset (ordered by

inclusion

), and

𝒪

⁢

(

P

)

∂

=

𝒪

⁢

(

P

∂

)

, where

∂

is the dualization

operation

(meaning that

P

∂

is the

dual poset

of

P

).

Title

upper set

Canonical name

UpperSet

Date of creation

2013-03-22 15:49:50

Last modified on

2013-03-22 15:49:50

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

20

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06A06

Synonym

up set

Synonym

down set

Synonym

upper closure

Synonym

lower closure

Related topic

LatticeIdeal

Related topic

LatticeFilter

Related topic

Filter

Defines

lower set

Defines

upper closed

Defines

lower closed

Generated on Fri Feb 9 09:53:06 2018 by

LaTeXML
upper set operation is a closure operator

upper set operation is a closure operator

In this entry, we shall prove the assertion made in
the main entry (

http://planetmath.org/UpperSet

) that

↑

is a

closure operator

. This will be done by checking
that the

defining properties

are satisfied. To begin,
recall the

definition

of our

operation

:

Definition 1

.

Let

P

be a poset and

A

a subset of

P

. The

upper set

of

A

is defined to be the set

↑

A

=

{

b

∈

P

∣

(

∃

a

∈

A

)

a

≤

b

}

Now, we verify each of the

properties

which is required
of a closure operator.

Theorem 1

.

↑

∅

=

∅

Proof.

Any statement of the form “

(

∃

a

∈

∅

)

P

(

a

)

”
is identically false no matter what the

predicate

P

(i.e. it is
an antitautology) and the set of objects satsfying an identically
false condition is empty, so

↑

∅

=

∅

.
∎

Theorem 2

.

A

⊆

↑

A

Proof.

This follows from

reflexivity

— for every

a

∈

A

, one has

a

≤

a

, hence

a

∈

↑

A

.
∎

Theorem 3

.

↑

↑

A

=

↑

A

Proof.

By the previous result,

↑

A

⊆

↑

↑

A

. Hence,
it only remains to show that

↑

↑

A

⊆

↑

A

.
This follows from

transitivity

. In order for some

a

to
be an element of

↑

↑

A

, there must exist

b

and

c

such that

a

≥

b

≥

C

and

C

∈

A

. By transitivity,

A

≥

C

,
so

a

∈

↑

A

, hence

↑

↑

A

⊆

↑

A

as well.
∎

Theorem 4

.

If

A

and

B

are subsets of a

partially ordered set

, then

↑

(

A

∪

B

)

=

(

↑

A

)

∪

(

↑

B

)

Proof.

On the one hand, if

a

∈

↑

(

A

∪

B

)

, then

a

≥

b

for
some

b

∈

A

∪

B

. It then follows that either

b

∈

A

or

b

∈

B

. In the former case,

a

∈

↑

A

, in the latter
case,

a

∈

↑

B

so, either way

a

∈

(

↑

A

)

∪

(

↑

B

)

.
Hence

↑

(

A

∪

B

)

⊆

(

↑

A

)

∪

(

↑

B

)

.

On the other hand, if

a

∈

(

↑

A

)

∪

(

↑

B

)

, then either

a

∈

(

↑

A

)

or

a

∈

(

↑

B

)

. In the former case, there
exists

b

such that

a

≥

b

and

b

∈

A

. Since

A

⊆

A

∪

B

, we also have

b

∈

A

∪

B

, hence

a

∈

↑

(

A

∪

B

)

. Likewise, in the second case, we also conclude
that

a

∈

↑

(

A

∪

B

)

. Therefore, we have

(

↑

A

)

∪

(

↑

B

)

⊆

↑

(

A

∪

B

)

.
∎

Theorem 5

.

↑

P

=

P

Theorem 6

.

A

⊆

B

,

↑

A

⊆

↑

B

Title

upper set operation is a closure operator

Canonical

name

UpperSetOperationIsAClosureOperator

Date of creation

2013-03-22 16:41:43

Last modified on

2013-03-22 16:41:43

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

8

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 06A06

Generated on Fri Feb 9 09:53:10 2018 by

LaTeXML
Zorn’s lemma

Zorn’s lemma

If

X

is a

partially ordered set

such that every chain in

X

has an

upper bound

,
then

X

has a

maximal element

.

Note that the empty chain in

X

has an upper bound in

X

if and only if

X

is non-empty.
Because this case is rather different from the case of non-empty chains,
Zorn’s Lemma is often stated in the following form:
If

X

is a non-empty partially ordered set
such that every non-empty chain in

X

has an upper bound,
then

X

has a maximal element.
(In other words: Any non-empty

inductively ordered

set has a maximal element.)

In ZF, Zorn’s Lemma is

equivalent

to the

Axiom of Choice

(

http://planetmath.org/AxiomOfChoice

).

Title

Zorn’s lemma

Canonical name

ZornsLemma

Date of creation

2013-03-22 12:09:04

Last modified on

2013-03-22 12:09:04

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

10

Author

yark (2760)

Entry type

Theorem

Classification

msc 06A06

Classification

msc 03E25

Related topic

AxiomOfChoice

Related topic

MaximalityPrinciple

Related topic

HaudorffsMaximumPrinciple

Related topic

ZornsLemmaAndTheWellOrderingTheoremEquivalenceOfHaudorffsMaximumPrinciple

Related topic

EveryVectorSpaceHasABasis

Related topic

TukeysLemma

Related topic

ZermelosPostulate

Related topic

KuratowskisLemma

Related topic

EveryRingHasAMaximalIdeal

Related topic

InductivelyOr

Generated on Fri Feb 9 09:53:12 2018 by

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