#noindex
'''순서,order''' VG: 순서,order x
----
기호

[[순서론,order_theory]]에선
 (일반적인 수학에서 크기를 비교하는) < ...를 쓰기도 하고
 휘어진 ≺ .... 를 쓰기도 한다. (이하 8개)
  ≺ WtEn:≺ U+227A PRECEDES
  ≻ WtEn:≻ U+227B SUCCEEDS
  ≼ WtEn:≼ U+227C PRECEDES OR EQUAL TO { precedes or equal to }
  ≽ WtEn:≽ U+227D SUCCEEDS OR EQUAL TO
  and
  ≾ WtEn:≾ U+227E PRECEDES OR EQUIVALENT TO
  ≿ WtEn:≿ U+227F SUCCEEDS OR EQUIVALENT TO
  ⊀ WtEn:⊀ U+2280 DOES NOT PRECEDE
  ⊁ WtEn:⊁ U+2281 DOES NOT SUCCEED
  ...
  TeX로는 https://www.access2science.com/latex/Binary.html 참조.

tmp bmks todel ... via Ggl:≺
https://math.stackexchange.com/questions/1614875/what-does-≼-or-≺-mean
https://en.wikipedia.org/wiki/Ordered_set_operators
----
밀접:
 인덱스
  순서를 가진 것 ~= [[인덱스,index]]를 가진 것
----
Sub:

ascending_order = increasing_order // WtEn:ascending_order ? WtEn:increasing_order ?
 ascending order / increasing order
 Twins:
 https://oeis.org/wiki/Ascending_order
  ( Redirected from https://oeis.org/wiki/Increasing_order )
 mkl/cmp: [[increasing_sequence]] (curr at [[시퀀스,sequence]])
 Up: ( increment - [[increment]] - 증가, [[증분,increment]] , [[growth]] )
descending_order // WtEn:descending_order ?
 descending order 
 Twins:
 https://oeis.org/wiki/Descending_order
 mkl/cmp: [[decreasing_sequence]] (curr at [[시퀀스,sequence]])
 Up: ( decrement - [[decrement]] - 감소, ..., [[decay]] )
[[순서공리]] (rename, [[순서공리,order_axiom]] 순서공리)
[[order_isomorphism]] =,order_isomorphism . order_isomorphism
 {
 order isomorphism
 순서동형사상
 
 order isomorphic adj. 순서동형(인) ?? chk
 순서동형

 [[순서,order]]
 [[동형사상,isomorphism]]

 https://en.wikipedia.org/wiki/Order_isomorphism
  monotone_function = monotonic_function 의 일종

 ... Ggl:순서동형

 trans comfirmed via KmsK:순서동형 => "order isomorphism	순서동형사상"
 WtEn:order_isomorphism x [[Date(2024-01-24T07:31:32)]]
 } // order isomorphism Ggl:"order isomorphism"
[[순서쌍,ordered_pair]] =순서쌍,ordered_pair =,ordered_pair 순서쌍 ordered_pair //OR//
[[순서짝,ordered_pair]] =순서짝,ordered_pair =,ordered_pair 순서짝 ordered_pair //TOO//
{
KmsK:순서쌍
{ as of [[Date(2023-10-03T15:51:45)]]
ordered pair	순서쌍, 순서짝
unordered pair	무순서쌍, 순서없는 쌍
}
Ndict:순서쌍

Srch:순서쌍 Srch:ordered_pair

KmsE:"ordered pair"
https://en.wiktionary.org/wiki/ordered_pair
// VG: 순서쌍,ordered_pair x
"ordered pair"
} // ordered pair 순서쌍/순서짝

[[순서위상,order_topology]]

[[순서론,order_theory]]

[[순서관계,order_relation]] - [[관계,relation]] ? 아님 순서가 곧 이항관계? ( Up: [[이항관계,binary_relation]]? )
{w
'''order relation'''
'''순서관계'''

https://www.gabormelli.com/RKB/Order_Relation

[[Namu:순서%20관계]]

[[https://encyclopediaofmath.org/wiki/Order_(on_a_set)]] - ''order relation''

} // order relation Ggl:"order relation" Ndict:순서관계 Ggl:순서관계

rel?
[[우선순위,priority]]

 * ascending order
 * descending order

/// logic에서의 order
{

[[일차논리,first-order_logic]]
 RR - curr at [[로직,logic?action=highlight&value=1st]]
 VG 일차논리,first-order_logic o


}

/// [[군,group]] [[군론,group_theory]]에서의 order.
{
[[https://artofproblemsolving.com/wiki/index.php/Order_(group_theory)]]
}
----
(수학 밖)
[[어순,word_order]] =어순,word_order =,word_order 어순 word_order ?
{
WtEn:word_order
https://namu.wiki/w/어순
}

[[z-order]] =,z-order . z-order
[[z순서,z-order]] ?
{ 
'''z-order'''

WtEn:z-order ?

[[computer_graphics]]쪽에서?
} // z-order Ggl:z-order

----
MKLINK
[[인덱스,index]]

----
<<tableofcontents>>

= ordinal (number) =
[[ordinal_number]] =,ordinal_number . ordinal_number
{
QQQ ordinal number 를 줄여서 ordinal 이라고도?

MKLINK
[[counting]] { [[counting_theorem]] { https://proofwiki.org/wiki/Counting_Theorem } }
[[cardinal]] [[cardinal_number]] { https://proofwiki.org/wiki/Definition:Cardinal https://proofwiki.org/wiki/Definition:Cardinal_Number }

WtEn:ordinal_number
https://proofwiki.org/wiki/Definition:Ordinal
}

= partial order, partially ordered set =
MKLINK
[[Zorn_lemma]]
[[Hasse_diagram]]


[[부분순서,partial_order]] =부분순서,partial_order =,partial_order 부분순서 partial_order
{

Sub:

complete_partial_order 
 완전부분순서 ? Ggl:완전부분순서 Naver:완전부분순서
 algebraic_complete_partial_order { http://www.pls-lab.org/en/Algebraic_complete_partial_order }

total_order = linear_order
전순서 선형순서 ?
Ggl:"전순서 선형순서 " Naver:"전순서 선형순서 "
부분순서에, 임의의 두 원소가 비교가능하다는 조건까지 추가된 더 엄격한 순서? chk


[[러스트,Rust]] API > trait `std::cmp::PartialOrd
https://doc.rust-lang.org/std/cmp/trait.PartialOrd.html

----
https://planetmath.org/partialorder
https://proofwiki.org/wiki/Definition:Partial_Ordering

Ndict:부분순서
부분순서
}

[[부분순서집합,partially_ordered_set]] OR
[[부분순서집합,partially_ordered_set,poset]]
=부분순서집합,partially_ordered_set,poset =,partially_ordered_set =,poset . 부분순서집합 partially_ordered_set poset
{
[[WtEn:partially_ordered_set]] = https://en.wiktionary.org/wiki/partially_ordered_set
[[WtEn:poset]] = https://en.wiktionary.org/wiki/poset


Sub: complately_partially_ordered_set ? { https://en.wikipedia.org/wiki/Complete_partial_order }


----
https://en.wikipedia.org/wiki/Partially_ordered_set

https://artofproblemsolving.com/wiki/index.php/Partially_ordered_set
https://encyclopediaofmath.org/wiki/Partially_ordered_set
https://proofwiki.org/wiki/Definition:Partially_Ordered_Set
}

= total order, totally ordered set =
전순서?
전체순서?

[[total_order]] =,total_order =,to .
{
total_order = linear_order
WtEn:total_order 
WtEn:linear_order

"is a partial_order in which any two elements are comparable"(we)

Sub:

strict_total_order 
strict_linear_order
non-strict_total_order 
non-strict_linear_order
''see https://en.wikipedia.org/wiki/Total_order#Strict_and_non-strict_total_orders''

[[러스트,Rust]] API > trait `std::cmp::ord`
https://doc.rust-lang.org/std/cmp/trait.Ord.html
Trait for types that form a total order.

Twin
https://en.wikipedia.org/wiki/Total_order
... Google:total.order
Up: [[partial_order]]
}

[[totally_ordered_set]]
{
https://encyclopediaofmath.org/wiki/Totally_ordered_set - ''chain, linear order'' // [[사슬,chain]] [[선형순서,linear_order]]
https://proofwiki.org/wiki/Definition:Totally_Ordered_Set
}

https://proofwiki.org/wiki/Definition:Total_Ordering

= well =

Kms:"well ord"

== well-ordering ==
https://proofwiki.org/wiki/Definition:Well-Ordering

=== well-ordering principle ===
well-ordering principle or well ordering principle
[[well-ordering_principle]] or [[well_ordering_principle]]

Twin

https://artofproblemsolving.com/wiki/index.php/Well_Ordering_Principle
 "every nonempty subset of the positive integers $\mathbb{Z}^{+}$ contains a smallest element."
 rel [[mathematical_induction]] ie [[귀납,induction]](of math)

...
"well ordering principle"
Ggl:"well ordering principle"

isa [[원리,principle]]

=== well-ordering theorem ===
[[well-ordering_theorem]] =,well-ordering_theorem =,well-ordering_theorem . well-ordering_theorem
well-ordering theorem


WtEn:well-ordering_theorem

https://en.wikipedia.org/wiki/Well-ordering_theorem
 "the well-ordering theorem, also known as Zermelo's theorem, states that every set can be well-ordered."

Ggl:"well-ordering theorem"
"well-ordering theorem"

== well-ordered set, woset ==
[[well-ordered_set]] =,well-ordered_set =,well-ordered_set . well-ordered_set
well-ordered set
https://encyclopediaofmath.org/wiki/Well-ordered_set
{
[[이항관계,binary_relation]] ≤와 집합과 네 조건으로 정의함.
}
https://proofwiki.org/wiki/Definition:Well-Ordered_Set
(Redirected from Definition:Woset ''i.e. https://proofwiki.org/wiki/Definition:Woset'')

Ggl:"well-ordered set"
Naver:"well-ordered set"
Bing:"well-ordered set"
"well-ordered set"

= lexicographic order =
[[lexicographic_order]]

https://encyclopediaofmath.org/wiki/Lexicographic_order
https://oeis.org/wiki/Orderings#Lexicographic_order

cmp [[alphabetical_order]]

= reverse lexicographic order =
[[reverse_lexicographic_order]]

https://oeis.org/wiki/Orderings#Reverse_lexicographic_order

= colexicographic, colexicographical order, colex order =
https://oeis.org/wiki/Orderings#Colexicographic_order

[#preorder]
= preorder #preorder =
원순서 ?
kms preorder => https://www.kms.or.kr/mathdict/list.html?key=ename&keyword=preorder

[[원순서,preorder]] - w

https://en.wiktionary.org/wiki/preorder

http://www.pls-lab.org/en/Preorder
 [[부분순서,partial_order]]와 거의 같지만 antisymmetry axiom을 만족시키지 않는.

https://planetmath.org/preorder

= pointwise order =
[[pointwise_order]]
점별순서
점마다순서

at [[Date(2023-08-08T11:00:06)]]
WtEn:pointwise_order - none.
WpEn:Pointwise_order - redir to WpEn:Pointwise#Pointwise_relations
Ndict:"pointwise order"
NdEn:"pointwise"
NdEn:pointwise*

Up: [[pointwise]]
{
Srch:pointwise
Prefix for mathematical terms?
// kms pointwise : '점마다, 점별' .... ~~https://www.kms.or.kr/mathdict/list.html?key=ename&keyword=pointwise~~ KmsE:pointwise
Cmp: [[piecewise]]
{
Srch:piecewise
// kms piecewise : 조각/조각마다/조각적 ... --https://www.kms.or.kr/mathdict/list.html?key=ename&keyword=piecewise-- KmsE:piecewise 
// kms wise : https://www.kms.or.kr/mathdict/list.html?key=ename&keyword=wise
.....
KmsE:piecewise 
KmsE:wise
}


Sub:
[[pointwise_order]]  =,pointwise_order . pointwise_order
{ pointwise order
pointwise+order
}
[[pointwise_relation]] =,pointwise_relation . pointwise_relation
{ pointwise relation
WpEn:Pointwise#Pointwise_relations
pointwise+relation
}
[[pointwise_operation]] =,pointwise_operation . pointwise_operation
{
... Google:pointwise+operation  Naver:pointwise+operation
}

pointwise}
pointwise+order ... Google:pointwise+order Naver:pointwise+order

= order structure =
order structure
[[order_structure]] =,order_structure =,order_structure . order_structure
=순서구조,order_structure ?  is a [[구조,structure]]?

WtEn:order_structure ?
WpSp:order_structure ?
WpEn:order_structure ?

https://en.wikipedia.org/wiki/List_of_order_structures_in_mathematics

Ggl:순서구조
Ggl:"order structure"

= alphabetical order =
[alphabetical_order]
WtEn:alphabetical_order
Ggl:"alphabetical order"

Rel [[알파벳,alphabet]]

= ADDHERE SUB =
= ADDHERE SUB =
= ADDHERE SUB =
= ADDHERE SUB =
= ADDHERE SUB =

= 같은 영단어 order의 다른 뜻 =
[[오더,order]]
...
RR:order

= 비슷? / 비교 =
Similar:
[[순위,rank]] or [[랭크,rank]]


----
http://emal.iptime.org/noriwiki/index.php/Order

https://oeis.org/wiki/Orderings

https://mathworld.wolfram.com/Ordering.html
 [[조합,combination]] [[순열,permutation]] (or [[치환,permutation]]) KmsE:permutation