aarjaneiro/planetmath huggingface dataset:
connected poset

connected poset

Let

P

be a poset. Write

a

⟂

b

if either

a

≤

b

or

b

≤

a

. In other words,

a

⟂

b

if

a

and

b

are

comparable

. A poset

P

is said to be

connected

if for every pair

a

,

b

∈

P

, there is a

finite sequence

a

=

c

1

,

c

2

,

…

,

c

n

=

b

, with each

c

i

∈

P

, such that

c

i

⟂

c

i

+

1

for each

i

=

1

,

2

,

…

,

n

-

1

.

For example, a poset with the property that any two

elements

are either

bounded from above

or bounded from below is a connected poset. In particular, every

semilattice

is connected. A fence is always connected. If

P

has more than one element and contains an element that is both

maximal

and

minimal

, then it is not connected. A

connected component

in a poset

P

is a maximal connected subposet. In the last example, the maximal-minimal point is a

component

in

P

. Any poset can be written as a

disjoint union

of its components.

It is true that a poset is connected if its corresponding

Hasse diagram

is a

connected graph

. However, the

converse

is not true. Before we see an example of this, let us recall how to construct a Hasse diagram from a poset

P

. The diagram so constructed is going to be an

undirected graph

(since this is all we need in our discussion). Draw an edge between

a

,

b

∈

P

if either

a

covers

b

or

b

covers

a

. Let us denote this

relation

between

a

and

b

by

a

≍

b

. Let

E

be the

collection

of all these edges. Then

G

=

(

P

,

E

)

is a graph where elements of

P

serve as vertices and

E

as the constructed edges. From this construction, one sees that a finite path exists between

a

,

b

∈

V

⁢

(

G

)

=

P

if there is a finite sequence

a

=

d

0

,

d

1

,

…

,

d

m

=

b

, with each

d

i

∈

V

⁢

(

G

)

, such that

d

i

≍

d

i

+

1

for

i

=

1

,

…

,

m

-

1

. In other words,

a

and

b

can be “joined” by a finite number of edges, such that

a

is a vertex on the first edge and

b

is the vertex on the last edge.

As promised, here is an example of a connected poset whose underlying Hasse diagram is not connected. take the real line

ℝ

with

∞

adjoined to the right (meaning every element

r

∈

ℝ

is less than or equal to

∞

). Then the resulting poset is connected, but its underlying Hasse diagram is not, since no element in

ℝ

can be joined to

∞

by a finite path.

Title

connected poset

Canonical name

ConnectedPoset

Date of creation

2013-03-22 17:08:31

Last modified on

2013-03-22 17:08:31

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06A07

Related topic

ConnectedGraph

Defines

connected

Defines

connected component

Generated on Fri Feb 9 09:53:24 2018 by

LaTeXML
Newtonian coalgebra

Newtonian coalgebra

Let

R

be a

commutative ring

. A

Newtonian coalgebra

over

R

is an

R

-module

C

which is simultaneously a coalgebra with comultiplication

Δ

:

C

→

C

⊗

C

and an

algebra

with

multiplication

⋅

:

C

⊗

C

→

C

such that

Δ

is a

derivation

over

⋅

, that is, such that the

identity

Δ

⁢

(

u

⋅

v

)

=

Δ

⁢

(

u

)

⋅

v

+

u

⋅

Δ

⁢

(

v

)

holds for any

u

and

v

in

C

. Newtonian coalgebras were introduced by Joni and Rota in

[

5

]

, where they were called

infinitesimal

coalgebras

. They reserved the term “Newtonian coalgebra” for the special case of the coalgebra of divided differences. This example was studied in more detail by Hirschhorn and Raphael

[

4

]

. Joni and Rota also showed that Newtonian coalgebras provide a

language

which can explain iterated differentiation of trigonometric functions as well as Faà di Bruno’s

formula

. See also the paper of Nichols and Sweedler

[

6

]

for more on trigonometric coalgebras.

A Newtonian coalgebra cannot have both a unit and a counit, so no Newtonian coalgebra is a

Hopf algebra

. However, Aguiar

[

1

]

developed a notion of antipode that makes sense for Newtonian coalgebras, leading to what he calls an infinitesimal Hopf algebra. Ehrenborg and Readdy

[

3

]

used Newtonian coalgebras to give an

algebraic structure

to the

𝐜𝐝

-index (

http://planetmath.org/CdIndex

), a poset

invariant

generalizing the

f

-vector of polytopes.

One example of a Newtonian coalgebra is the

free associative algebra

R

⁢

⟨

𝐚

,

𝐛

⟩

of

polynomials

on the noncommuting

variables

𝐚

and

𝐛

with

coefficients

in

R

. The

product

is the ordinary

noncommutative

polynomial product, and the comultiplication is defined by setting

Δ

⁢

(

u

1

⁢

⋯

⁢

u

n

)

=

∑

j

∈

[

n

]

u

1

⁢

⋯

⁢

u

i

-

1

⊗

u

i

+

1

⁢

⋯

⁢

u

n

for each

monomial

and extending by linearity.

References

1

M. Aguiar, Infinitesimal Hopf algebras.

New trends in Hopf algebra theory: (La Falda, 1999)

, 1–29, Contemp. Math., 267, Amer. Math. Soc., Providence, RI, 2000.

2

M. Aguiar, Infinitesimal Hopf algebras and the

𝐜𝐝

-index of polytopes.

Discrete Comput. Geom.

, 27 (2002), no. 1, 3–28.

3

R. Ehrenborg and M. Readdy,

Coproducts

and the

𝐜𝐝

-index,

J. Algebr. Comb.

, 8 (1998), 273–299.

4

P. S. Hirschhorn and L. A. Raphael, Coalgebraic

foundation

of the method of divided differences,

Adv. Math.

, 91 (1992), 75–135.

5

S. A. Joni and G.-C. Rota,

Coalgebras and

bialgebras

in

combinatorics

, Stud. Appl. Math., 61 (1979), pp. 93–139.

6

W. Nichols and M. Sweedler,

Hopf algebras and combinatorics

, in

Proceedings of the conference on

umbral calculus

and Hopf algebras

, ed. R. Morris, AMS, 1982.

Title

Newtonian coalgebra

Canonical name

NewtonianCoalgebra

Date of creation

2013-03-22 16:48:36

Last modified on

2013-03-22 16:48:36

Owner

mps (409)

Last modified by

mps (409)

Numerical id

6

Author

mps (409)

Entry type

Definition

Classification

msc 06A11

Classification

msc 16W30

Generated on Fri Feb 9 09:53:35 2018 by

LaTeXML
rank-selected poset

rank-selected poset

Let

P

be a

graded poset

of rank

n

+

1

with rank function

ρ

. For any

S

⊂

{

0

,

1

,

…

,

n

+

1

}

let

P

S

denote the subset

P

S

=

{

x

∈

P

:

ρ

⁢

(

x

)

∈

S

}

=

ρ

-

1

⁢

(

S

)

.

Each such subset inherits a poset

structure

from

P

as an

induced

poset. So we call

P

S

the

rank-selected poset

of

P

induced by

S

, or more briefly the

S

-rank-selected subposet of

P

.

The rank-selected posets of a poset

P

can be used to define two special

arithmetic

invariants

of

P

. First for each

S

, the

alpha invariant

α

S

⁢

(

P

)

is the number of saturated chains in

P

S

. Then define

β

S

⁢

(

P

)

by

β

S

⁢

(

P

)

=

∑

T

⊂

S

(

-

1

)

|

S

|

-

|

T

|

⁢

α

T

⁢

(

P

)

.

The invariant

β

is called the

rank-selected Möbius invariant

of

P

.

For example, let

L

be the face poset of a

convex polytope

P

of

dimension

n

, including the special

elements

0

^

(representing the empty face) and

1

^

(representing the interior of the

polytope

). For any

i

∈

{

0

,

…

,

n

-

1

}

, the alpha invariant

α

{

i

+

1

}

⁢

(

L

)

counts the number of faces of

P

of dimension

i

. For arbitrary

S

⊂

{

1

,

…

,

n

}

, the numbers

α

S

⁢

(

L

)

are entries in the flag

f

-vector of

P

and thus count flags of faces in

P

, while the

β

S

⁢

(

L

)

are entries in the flag

h

-vector of

P

.

While the alpha invariant is by construction always nonnegative, the Möbius invariant is not guaranteed to be nonnegative. Posets for which the Möbius invariant is always nonnegative (and therefore counts something) are of special interest to combinatorialists. In particular, the Möbius invariant is nonnegative for face posets of convex polytopes.

References

1

Stanley, R.,

Enumerative Combinatorics

, vol. 1, 2nd ed., Cambridge
University Press, Cambridge, 1996.

Title

rank-selected poset

Canonical name

RankselectedPoset

Date of creation

2013-03-22 16:23:43

Last modified on

2013-03-22 16:23:43

Owner

mps (409)

Last modified by

mps (409)

Numerical id

5

Author

mps (409)

Entry type

Definition

Classification

msc 06A11

Classification

msc 06A06

Defines

alpha invariant

Defines

beta invariant

Defines

rank-selected Möbius invariant

Defines

rank-selected Mobius invariant

Generated on Fri Feb 9 09:53:38 2018 by

LaTeXML
bounded complete

bounded complete

Let

P

be a poset. Recall that a subset

S

of

P

is called

bounded from above

if there is an

element

a

∈

P

such that, for every

s

∈

S

,

s

≤

a

.

A poset

P

is said to be

bounded complete

if every subset which is bounded from above has a

supremum

.

Remark

. Since it is not required that the subset be non-empty, we see that

P

has a bottom. This is because the

empty set

is

vacuously

bounded from above, and therefore has a supremum. However, this supremum is less than or equal to every member of

P

, and hence it is the least element of

P

.

Clearly, any

complete lattice

is bounded complete. An example of a non-complete bounded complete poset is any closed subset of

ℝ

of the form

[

a

,

∞

)

, where

a

∈

ℝ

. In

addition

, arbitrary

products

of bounded complete posets is also bounded complete.

It can be shown that a poset is a bounded complete dcpo iff it is a

complete semilattice

.

Remark

. A weaker

concept

is that of

Dedekind completeness

: A poset

P

is

Dedekind complete

if every

non-empty

subset bounded from above has a supremum. An

obvious

example is

ℝ

, which is Dedekind complete but not bounded complete (as it has no bottom). Dedekind completeness is more commonly known as the least upper bound

property

.

Title

bounded complete

Canonical

name

BoundedComplete

Date of creation

2013-03-22 17:01:08

Last modified on

2013-03-22 17:01:08

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06A12

Classification

msc 06B23

Classification

msc 03G10

Related topic

CompletenessPrinciple

Defines

Dedekind complete

Generated on Fri Feb 9 09:53:47 2018 by

LaTeXML
join

join

Certain posets

X

have a

binary operation

join

denoted by

∨

, such that

x

∨

y

is the least

upper bound

of

x

and

y

. Such posets are called

join-semilattices

, or

∨

-semilattices

, or

upper semilattices

.

If

j

and

j

′

are both joins of

x

and

y

, then

j

≤

j

′

and

j

′

≤

j

, and so

j

=

j

′

; thus a join, if it exists, is unique. The join is also known as the

or operator

.

Title

join

Canonical

name

Join

Date of creation

2013-03-22 12:27:40

Last modified on

2013-03-22 12:27:40

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

11

Author

yark (2760)

Entry type

Definition

Classification

msc 06A12

Synonym

or operator

Related topic

Meet

Related topic

Semilattice

Defines

join-semilattice

Defines

join semilattice

Defines

upper semilattice

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

LaTeXML
meet

meet

Certain posets

X

have a

binary operation

meet

denoted by

∧

, such that

x

∧

y

is the

greatest lower bound

of

x

and

y

. Such posets are called

meet-semilattices

, or

∧

-semilattices

, or

lower semilattices

.

If

m

and

m

′

are both meets of

x

and

y

, then

m

≤

m

′

and

m

≥

m

′

, and so

m

=

m

′

; thus a meet, if it exists, is unique. The meet is also known as the

and

operator

.

Title

meet

Canonical

name

Meet

Date of creation

2013-03-22 12:27:37

Last modified on

2013-03-22 12:27:37

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

10

Author

yark (2760)

Entry type

Definition

Classification

msc 06A12

Synonym

and operator

Related topic

Join

Related topic

Semilattice

Defines

meet-semilattice

Defines

meet semilattice

Defines

lower semilattice

Generated on Fri Feb 9 09:54:11 2018 by

LaTeXML
meet continuous

meet continuous

Let

L

be a

meet semilattice

. We say that

L

is

meet continuous

if

1.

for any

monotone

net

D

=

{

x

i

∣

i

∈

I

}

in

L

, its

supremum

⋁

D

exists, and

2.

for any

a

∈

L

and any monotone net

{

x

i

∣

i

∈

I

}

,

a

∧

⋁

{

x

i

∣

i

∈

I

}

=

⋁

{

a

∧

x

i

∣

i

∈

I

}

.

A monotone net

{

x

i

∣

i

∈

I

}

is a net

x

:

I

→

L

such that

x

is a non-decreasing

function

; that is, for any

i

≤

j

in

I

,

x

i

≤

x

j

in

L

.

Note that we could have replaced the first condition by saying simply that

D

⊆

L

is a

directed set

. (A monotone net is a directed set, and a directed set is a trivially a monotone net, by considering the

identity function

as the net). It’s not hard to see that if

D

is a directed subset of

L

, then

a

∧

D

:=

{

a

∧

x

∣

x

∈

D

}

is also directed, so that the right hand side of the second condition makes sense.

Dually, a

join semilattice

L

is

join continuous

if its dual (as a meet semilattice) is meet continuous. In other words, for any

antitone

net

D

=

{

x

i

∣

i

∈

I

}

, its

infimum

⋀

D

exists and that

a

∨

⋀

{

x

i

∣

i

∈

I

}

=

⋀

{

a

∨

x

i

∣

i

∈

I

}

.

An antitone net is just a net

x

:

I

→

L

such that for

i

≤

j

in

I

,

x

j

≤

x

i

in

L

.

Remarks

.

•

A meet continuous

lattice

is a

complete lattice

, since a poset such that finite joins and directed joins exist is a complete lattice (see the link below for a proof of this).

•

Let a lattice

L

be both meet continuous and join continuous. Let

{

x

i

∣

i

∈

I

}

be any net in

L

. We define the following:

lim

¯

⁢

x

i

=

⋀

j

∈

I

{

⋁

j

≤

i

x

i

}

and

lim

¯

⁢

x

i

=

⋁

j

∈

I

{

⋀

i

≤

j

x

i

}

If there is an

x

∈

L

such that

lim

¯

⁢

x

i

=

x

=

lim

¯

⁢

x

i

, then we say that the net

{

x

i

}

order converges

to

x

, and we write

x

i

→

x

, or

x

=

lim

⁡

x

i

. Now, define a subset

C

⊆

L

to be

closed

(in

L

) if for any net

{

x

i

}

in

C

such that

x

i

→

x

implies that

x

∈

C

, and

open

if its set

complement

is closed, then

L

becomes a

topological lattice

. With respect to this

topology

, meet

∧

and join

∨

are easily seen to be

continuous

.

References

1

G. Birkhoff,

Lattice Theory

, 3rd Edition, Volume 25, AMS, Providence (1967).

2

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).

3

G. Grätzer,

General Lattice Theory

, 2nd Edition, Birkhäuser (1998).

Title

meet continuous

Canonical name

MeetContinuous

Date of creation

2013-03-22 16:36:41

Last modified on

2013-03-22 16:36:41

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

12

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06A12

Classification

msc 06B35

Synonym

order convergence

Related topic

CriteriaForAPosetToBeACompleteLattice

Related topic

JoinInfiniteDistributive

Defines

join continuous

Defines

order converges

Generated on Fri Feb 9 09:54:21 2018 by

LaTeXML
ascending chain condition

ascending chain condition

A

partially ordered set

S

(for example, a collection of subsets of a set

X

when ordered by inclusion) satisfies the

ascending chain condition

or

ACC

if there does not exist an infinite ascending chain

s

1

<

s

2

<

⋯

of elements of

S

.

See also the

descending chain condition

(DCC).

Title

ascending chain condition

Canonical name

AscendingChainCondition

Date of creation

2013-03-22 12:01:12

Last modified on

2013-03-22 12:01:12

Owner

antizeus (11)

Last modified by

antizeus (11)

Numerical id

8

Author

antizeus (11)

Entry type

Definition

Classification

msc 06A99

Synonym

ACC

Related topic

FactorChainCondition

Generated on Fri Feb 9 09:54:28 2018 by

LaTeXML
ascending order

ascending order

A

sequence

or arbitrary

ordered set

or one-dimensional array of numbers,

a

, is said to be in

ascending order

if each

a

i

≤

a

i

+

1

. For example, the

Fibonacci sequence

is in ascending order: 1, 1, 2, 3, 5, 8, 13, 21 … The

Perrin sequence

is not in ascending order: 3, 0, 2, 3, 2, 5, 5, 7, 10, 12, 17 …

In a trivial sense, the sequence of values of the sign function is in ascending order: … -1, -1, -1, 0, 1, 1, 1… When each

a

i

<

a

i

+

1

in the sequence, set or array, then it can be said to be in

strictly ascending order

.

Title

ascending order

Canonical name

AscendingOrder

Date of creation

2013-03-22 16:06:39

Last modified on

2013-03-22 16:06:39

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

7

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 06A99

Related topic

DescendingOrder

Defines

strictly ascending order

Generated on Fri Feb 9 09:54:30 2018 by

LaTeXML
dense (in a poset)

dense (in a poset)

If

(

P

,

≤

)

is a poset then a subset

Q

⊆

P

is dense if for any

p

∈

P

there is some

q

∈

Q

such that

q

≤

p

.

Title

dense (in a poset)

Canonical

name

DenseinAPoset

Date of creation

2013-03-22 12:55:58

Last modified on

2013-03-22 12:55:58

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

4

Author

Henry (455)

Entry type

Definition

Classification

msc 06A99

Related topic

Dense

Generated on Fri Feb 9 09:54:42 2018 by

LaTeXML
descending chain condition

descending chain condition

A

partially ordered set

S

(for example, a

collection

of subsets of a set

X

, ordered by

inclusion

)

satisfies

the

descending chain condition

or

DCC

if there does not exist an infinite descending chain

s

1

>

s

2

>

⋯

of elements of

S

.

See also the

ascending chain condition

(ACC).

Title

descending chain condition

Canonical name

DescendingChainCondition

Date of creation

2013-03-22 12:01:14

Last modified on

2013-03-22 12:01:14

Owner

antizeus (11)

Last modified by

antizeus (11)

Numerical id

7

Author

antizeus (11)

Entry type

Definition

Classification

msc 06A99

Synonym

DCC

Generated on Fri Feb 9 09:54:44 2018 by

LaTeXML
descending order

descending order

A

sequence

or arbitrary

ordered set

or one-dimensional array of numbers,

a

, is said to be in

descending order

if each

a

i

≥

a

i

+

1

. For example, the

aliquot sequence

of 259 is in descending order: 45, 33, 15, 9, 4, 3, 1, 0, 0, 0 … The aliquot sequence starting at 60, however, is not in descending order: 108, 172, 136, 134, 70, 74, 40, 50, 43, 1, 0, 0, 0 …

In a trivial sense, the sequence of values of the sign function multiplied by -1 is in descending order: … 1, 1, 1, 0, –1, –1, –1… When each

a

i

>

a

i

+

1

in the sequence, set or array, then it can be said to be in

strictly descending order

.

Title

descending order

Canonical name

DescendingOrder

Date of creation

2013-03-22 16:06:49

Last modified on

2013-03-22 16:06:49

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

5

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 06A99

Related topic

AscendingOrder

Defines

strictly descending order

Generated on Fri Feb 9 09:54:45 2018 by

LaTeXML
inductively ordered

inductively ordered

A

partially ordered set

A

is

inductively ordered

iff every chain of elements of

A

has an

upper bound

in

A

.

Examples.

The

power set

2

M

of any set

M

is inductively ordered by the

set inclusion

(

http://planetmath.org/Set

); any

finite set

of

integers

is inductively ordered by

divisibility

.

Cf.

inductive set

.

Title

inductively ordered

Canonical name

InductivelyOrdered

Date of creation

2013-03-22 14:55:21

Last modified on

2013-03-22 14:55:21

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

8

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 06A99

Related topic

ZornsLemma

Defines

inductive order

Defines

inductively orderes set

Generated on Fri Feb 9 09:54:47 2018 by

LaTeXML
lexicographic order

lexicographic order

Let

A

be a set equipped with a

total order

<

, and let

A

n

=

A

×

⋯

×

A

be the

n

-fold

Cartesian product

of

A

. Then the

lexicographic order

<

on

A

n

is defined as follows:

If

a

=

(

a

1

,

…

,

a

n

)

∈

A

n

and

b

=

(

b

1

,

…

,

b

n

)

∈

A

n

,
then

a

<

b

if

a

1

<

b

1

or

a

1

=

b

1

,

⋮

a

k

=

b

k

,

a

k

+

1

<

b

k

+

1

for some

k

=

1

,

…

,

n

-

1

.

Examples

•

The lexicographic order yields a total order on the field of

complex numbers

.

•

The lexicographic order of words of

finite length

consisting of letters

′

′

(space)

<

a

<

b

<

⋯

<

y

<

z

is the

dictionary order

. To compare words of different length, one simply pads the shorter with

′

′

s from the right. For example,

prove

<

proved

<

proven

.

Properties

•

The lexicographic order is a total order.

•

If the original set is

well-ordered

, the

lexicographic ordering

on the product is also a well-ordering.

Title

lexicographic order

Canonical name

LexicographicOrder

Date of creation

2013-03-22 15:14:05

Last modified on

2013-03-22 15:14:05

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

13

Author

matte (1858)

Entry type

Definition

Classification

msc 06A99

Defines

dictionary order

Generated on Fri Feb 9 09:54:49 2018 by

LaTeXML
locally finite poset

locally finite poset

A poset

P

is

locally finite

if every

interval

[

x

,

y

]

in

P

is finite. For example,

ℤ

with the usual order is locally finite but not finite, while

ℚ

is neither.

Every locally finite poset is also

chain finite

, but the

converse

does not hold. To see this, define a

partial order

on

ℕ

by the rule that

k

≤

ℓ

if and only if

k

=

0

or

ℓ

=

1

. Thus

0

is the minimum

element

,

1

is the maximum element, and the remaining elements form an

infinite

antichain

. Every

bounded

chain in this poset is finite but the entire poset is an infinite interval, so the poset is chain finite but not locally finite.

Title

locally finite poset

Canonical name

LocallyFinitePoset

Date of creation

2013-03-22 14:09:15

Last modified on

2013-03-22 14:09:15

Owner

mps (409)

Last modified by

mps (409)

Numerical id

5

Author

mps (409)

Entry type

Definition

Classification

msc 06A99

Defines

locally finite

Generated on Fri Feb 9 09:54:50 2018 by

LaTeXML
pre-order

pre-order

Definition

A

pre-order

on a set

S

is a

relation

≲

on

S

satisfying the following two axioms:

reflexivity

:

s

≲

s

for all

s

∈

S

, and

transitivity

: If

s

≲

t

and

t

≲

u

, then

s

≲

u

; for all

s

,

t

,

u

∈

S

.

Partial order induced by a pre-order

Given such a relation, define a new relation

s

∼

t

on

S

by

s

∼

t

⁢

if and only if

⁢

s

≲

t

⁢

and

⁢

t

≲

s

.

Then

∼

is an

equivalence relation

on

S

, and

≲

induces a

partial order

≤

on the set

S

/

∼

of

equivalence classes

of

∼

defined by

[

s

]

≤

[

t

]

⁢

if and only if

⁢

s

≲

t

,

where

[

s

]

and

[

t

]

denote the equivalence classes of

s

and

t

. In particular,

≤

does

satisfy

antisymmetry, whereas

≲

may not.

Pre-orders as categories

A pre-order

≲

on a set

S

can be considered as a

small category

, in the which the objects are the elements of

S

and there is a unique

morphism

from

x

to

y

if

x

≲

y

(and none otherwise).

Title

pre-order

Canonical name

Preorder

Date of creation

2013-03-22 13:05:06

Last modified on

2013-03-22 13:05:06

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

17

Author

yark (2760)

Entry type

Definition

Classification

msc 06A99

Synonym

pre-ordering

Synonym

preorder

Synonym

preordering

Synonym

quasi-order

Synonym

quasi-ordering

Synonym

quasiorder

Synonym

quasiordering

Synonym

semi-order

Synonym

semi-ordering

Synonym

semiorder

Synonym

semiordering

Related topic

WellQuasiOrdering

Related topic

PartialOrder

Defines

pre-ordered

Defines

preordered

Defines

semi-ordered

Defines

semiordered

Defines

quasi-ordered

Defines

quasiordered

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

LaTeXML