[set theory] order relation (the relation between elements of partial order set | comparable | strictly less than | covering | Haas diagram)

One 、 Comparable

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Comparable :

A

A

A aggregate , There are Partial order relation

≼

\preccurlyeq

≼ Less than or equal to ,

Posets yes aggregate and Partial order relation The order of composition is right

<

A

,

≼

>

<A, \preccurlyeq>

<A,≼> ,

x

,

y

x, y

x,y yes

A

A

A Two elements in the set ,

x

,

y

∈

A

x , y \in A

x,y∈A ,

Or

x

≼

y

x \preccurlyeq y

x≼y , Either it's

y

≼

x

y \preccurlyeq x

y≼x , The symbolic representation is

x

≼

y

∨

y

≼

x

x \preccurlyeq y \lor y \preccurlyeq x

x≼y∨y≼x , You must choose one of the two situations ,

said

x

x

x And

y

y

y It's comparable ;

as long as

x

,

y

x, y

x,y Between There is a partial order relationship , No matter who is in front , Who's behind , all Unified name

x

x

x And

y

y

y It's comparable ;

Two 、 Strictly less than

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Strictly less than The concept needs to be based on Comparable concept

Strictly less than :

A

A

A aggregate And

A

A

A Upper partial order relation

≼

\preccurlyeq

≼ , form Posets

<

A

,

≼

>

<A, \preccurlyeq>

<A,≼> ,

x

,

y

x, y

x,y yes

A

A

A Two elements in the set ,

x

,

y

∈

A

x , y \in A

x,y∈A ,

If

x

,

y

x , y

x,y It's comparable (

x

,

y

x,y

x,y There is a partial order relationship between them ) , however

x

x

x And

y

y

y It's not equal , said

x

x

x Strictly less than

y

y

y ;

Symbolize :

x

≼

y

∧

x

≠

y

⇔

x

≺

y

x \preccurlyeq y \land x \not= y \Leftrightarrow x \prec y

x≼y∧x​=y⇔x≺y

3、 ... and 、 Cover

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Cover The concept needs to be based on Strictly less than concept

Cover :

A

A

A aggregate And

A

A

A Upper partial order relation

≼

\preccurlyeq

≼ , form Posets

<

A

,

≼

>

<A, \preccurlyeq>

<A,≼> ,

x

,

y

,

z

x, y , z

x,y,z yes

A

A

A The elements in the collection ,

x

,

y

,

z

∈

A

x , y , z \in A

x,y,z∈A ,

x

x

x Strictly less than

y

y

y ,

x

≺

y

x \prec y

x≺y ,

non-existent

z

z

z , send

x

x

x Strictly less than

z

z

z , also

z

z

z Strictly less than

y

y

y ,

said

y

y

y Cover

x

x

x ; ( Note that Big Cover Small )

In partial order relation Big Cover Small

Symbolize :

x

≺

y

∧

¬

∃

z

(

z

∈

A

∧

x

≺

y

≺

z

)

x \prec y \land \lnot \exist z( z \in A \land x \prec y \prec z )

x≺y∧¬∃z(z∈A∧x≺y≺z)

Four 、 Hastur

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A

A

A aggregate And

A

A

A Upper partial order relation

≼

\preccurlyeq

≼ , form Posets

<

A

,

≼

>

<A, \preccurlyeq>

<A,≼> ,

x

,

y

x, y

x,y yes

A

A

A Two elements in the set ,

x

,

y

∈

A

x , y \in A

x,y∈A ,

Hastur :

① The vertices : Use The vertices Express

A

A

A The elements in the collection ;

② No to the edge : If and only if

y

y

y Cover

x

x

x when ,

y

y

y The apex is at

x

x

x The vertices upper , And in

x

x

x The vertices And

y

y

y Between vertices Draw a No to the edge ;

Above,

6

6

6 Meta set The partial order relations on the

≼

\preccurlyeq

≼

A

A

A Element ratio

B

,

C

,

D

B,C,D

B,C,D Elements are small

Partial order relation is transitive ,

A

A

A Than

B

B

B Small ,

B

B

B Than

F

F

F Small , therefore

A

A

A Than

F

F

F Small

The bottom element

A

A

A The smallest is the smallest , All elements are better than

A

A

A Big ( Include

A

A

A , Partial order relation is reflexive )

Top element

F

F

F It's the biggest , All elements are better than

F

F

F Small ( Include

F

F

F , Partial order relation is reflexive )

B

C

D

E

BCDE

BCDE The four elements are not comparable to each other

Hastur And Diagram comparison Omitted content :

① Ring : Partial order relation is reflexive , therefore Every vertex has a ring , You can omit the ring

② arrow : Partial order relation is antisymmetric , therefore There must be no two-way edge between two vertices , They are all one-way edges , Therefore, the arrow direction can be omitted

③ Default direction : Use the up and down position to indicate the direction of the arrow , The arrow is up by default , Partial order is Less than or equal to , The smallest is on the smallest side , The biggest one is on the top ;