recursal/Fanatic-Fandom huggingface dataset:
Dr. Halsey

Dr. Halsey is an insane scientist and one of the leaders of the Empire of Chaturn.

## Description

Dr. Halsey is the head scientist of Chaturn and one of its leaders, seen issuing orders to characters like Doom Guy. She created the traitorous Spartan 117 and famously failed to recruit the Sun, instead being told to "fuck off" and crashing into Mercury. She also approved the construction of the Death Star which exploded due to problems with its gravity well, killing everyone on Mercury.

Due to her frequent incompetence and lack of sanity, she is not considered a special resource of Chaturn.

## History

### AI Invasion

Dr. Halsey led Chaturn's scientists in the project to create a clone of Master Chiefcalled Spartan 117, which immediately betrayed the Empire and joined their enemy, the Humans of Earth.

She went alone in her spacecraft to attempt to recruit the Sun, but the Sun told her to fuck off. Shocked by this, she immediately made a wrong turn and crashed into the surface of Mercury, but survived and escaped.

When the Empire recruited Doom Guy, Dr. Halsey issued the order for him to kill the Sun. She also approved the creation of a Death Star which immediately exploded and destroyed all life on Mercury.

As the lead scientist of Chaturn, Dr. Halsey presumably led the project to create Sun Tzu, which is small, ugly, and mean, in defiance of Empress Ghost's wishes for it to be big, beautiful, and nice.

After the Dougtopian Empire created Sun 3 by harvesting energy from Sun Tzu, Dr. Halsey flew over to Sun 3 and told it to "fuck off," still angry about the time the first Sun told her to fuck off. This caused the Dougtopian Empire to lose control of Sun 3, which was a win for Chaturn, their enemy.

When Chaturn acquired the planet of Uranus and renamed it to Ouranus, Chaturn's top scientists began construction of a Mega Toilet. In the process, they also invented nanobots, which some of the scientists injected into themselves, becoming super-intelligent machines called the Chatronomicon. It is unknown if Dr. Halsey was among the scientists who became the Chatronomicon, but it is not unlikely considering her insane nature. The scientists then created a Mega Robot Suit, which was never really used by Chaturn, but it sounds cool.

The top scientists, including the Chatronomicon, then built a new moon for Ouranus made out of cheese, called Moon Tzu. They also figured out how to create life from cheese, creating little animals made out of cheese called Cheese Creatures. They then finally finished the Mega Toilet on top of Moon Tzu, which was the size of Ouranus.
Hildr

**Hildr** is a Valkyrie introduced in The Reggiesworld Trilogy. She is one of the nine Valkyries who have been cursed by Odin to remain in a corporal form, eventually driving them to insanity. As she became too dangerous, Hildr eventually found herself sealed away by the Valkyrie Queen in Niflheim. She is nevertheless freed by Kratos and Atreus after they engaged and defeated her physical form. Blessing them with her eternal gratitude, her spirit vanished and eventually reunited with her sisters.

Hildr returns in *Ragnarök* as a Shield Maiden, having regained a pure physical form thanks to Freya and her Vanir magic. Grateful for having been freed from Odin's curse and having acknowledged Kratos as her General, she actively participates together with her freed sisters to the final assault against Asgard during Ragnarök.

## Trivia

* In legend, Hildr is the Mistress of Battle. Unlike her sisters, whom she doesn't get along with, Hildr got along quite well with the Allfather. Hildr would spend most of her time in Midgard, observing discord between the living and causing some herself at times. As she lived for conflict, many thought she was conflict personified.
* In retrospect as of *Ragnarök*, Hildr's fate of being among the Valkyries cursed by Odin became a rather curious case; Mimir described her as being close to the All-father and actually distant from the other Valkyries, yet she did not end up among Gná, Hrist and Mist who were loyal to Odin by choice and were spared from his maddening curse. Her reason to supposedly taking side against Odin has yet to be explained.
	+ A probable explanation is that Hildr was actually spying for the other 8 malcontent Valkyries; She did in fact, have a seat in the hidden Valkyrie council in Midgard, which deliberately matched the number of sisters that opposed Odin; the fact that she got cursed along her 8 sisters who later held no grudge or suspicion against her after they became Shield Maidens further reinforce this probability.
Lean River

Incredible

Wow

This is the origin of all leans, such as rare leans, space leans, magic leans, and crystalmeth leans. Its location is currently unknown. (Perhaps in Reggiesworld Universe)

The God of All Powers and the God of Animals are commonly seen here with ice cream. It is confirmed that the God of Cum used to live there and fish with his pet raccoon. The Lean River can be accessed through the Zalgo Ritual.

It is unknown where the lean comes from, but it is speculated that the lean participates in a cycle similar to the water cycle.

Cum-Mayo boxes can be found at lakeside outposts. At The Outpost, there are hillbillies who give away Powder Lean

Some areas along the river have leanberry trees. Sometimes pizza crust can be found.
KAM

Kingdom Apostolic Ministires or more commonly referred to as KAM is located in areas known as campuses. There are 4 current campuses. These campuses are Indianapolis, Detroit, Marion, and Virtual.

**Mt. Zion Apostolic Church Origin**

KAM was first founded by Pastor Lillian V. Mason in 1922 on the South Side of Indianapolis. The Church was named Christ Temple #2, and was originally in her house for services and prayer. In 1955, Pastor Mason directed the construction of the 2 story brick building which served the Church for 38 years. However on November 19th, 1967 Bishop James O. Franklin became the 2nd Pastor of Christ Temple #2. In 1969 Bishop Franklin helped facilitate the name change of Christ Temple #2 to Mt. Zion Apostolic Church. In 1978 Bishop Franklin also facilitated an addition to the Church building. On December 11, 1988 Bishop Lambert W. Gates, Sr. was installed as the 3rd Pastor of Mt. Zion. On May 4th, 1997 the Church marched into its new 38th street home. In June 2005 a new 1600 seating Sanctuary was added to the existing building.

**Greater Apostolic Faith Temple**

In April 1972 Bishop Smith (TBA)

***Unification***

In October 2007, Bishop Gates became the Shepard to both now sister churches. On the Watch-night service of 2020-21 this Unification came to be under 1 name instead of 2 which is Kingdom Apostolic Ministries

1. ↑ https://www.kam.church/
2. ↑ https://www.youtube.com/live/DfhE9cxinvs?feature=share&t=2553
3. ↑ https://www.youtube.com/live/uKwJZ8Wvr\_A?feature=share&t=8573
God’s

**Gods** are a race of powerful, ageless beings that are commonly worshiped by mortal inhabitants of certain realms. There are several, distinct pantheons of gods, such as the Greek Olympian’s or the Norse Aesir and Vanir pantheons. Every god in a pantheon fills a certain role, such as the God of War or the God of Light.

Oftentimes, each God has a counterpart in other pantheons. For instance, Týr is the Aesir God of War in the Norse pantheon. However, in the Greek pantheon, the God of War was Ares, who was later replaced by Kratos after the latter killed him.  According to Kratos, the gods care nothing for their worshipers. Worshiping them is foolish and men should not pray to monsters, however, Tyr is a notable exception as he is shown to greatly care about mortals.

Despite being immortal and immune to all forms of mortal harm, diseases and aging, Gods can still be killed, but only by the power of a God, either by another god or someone wielding a godly weapon. Kratos kills the entire Greek Pantheon in his quest for revenge against his father, Zeus, and thus leaves all of the Greek world in Chaos by the end of his vengeance.
Esben

**Esben** is the grandfather of Dwerve.

## Throughout *Dwerve*

Since Esben heard that Dwerve wants to go and fight in the Snoring Forest, Esben decided to teach him a few skills, such as how to place a Spinblade Turret towards dummy enemies.

After the training, Dwerve was told to go to his brewery. He does so by traveling to the overlook cliff while killing some enemies, then to the brewery to kill more enemies.

After he was about to return from the brewery, he saw that the Brekka Village was on fire. He was told by his grandfather not to go, but it felt like he had to because he wanted to save his father.

## Family

* Dwerve (grandson)
* Arik (son)
* Unnamed mother (daughter-in-law, deceased)
Set relation

**集合关系**基本术语主题。

术语
--

subset
nc.
**子集**
对两个集合

S

{\displaystyle S}

和

T

{\displaystyle T}

，如果

∀
x
∈
S
,
x
∈
T

{\displaystyle \forall x \in S, x \in T}

，我们就说集合

S

{\displaystyle S}

含于

T

{\displaystyle T}

，或

T

{\displaystyle T}

包含

S

{\displaystyle S}

，记作

S
⊆
T

{\displaystyle S \subseteq T}

或

T
⊇
S

{\displaystyle T \supseteq S}

，同时我们称

S

{\displaystyle S}

是

T

{\displaystyle T}

的**子集**
[例句]

equal
adj.
**相等**
如果

S
⊆
T
,
T
⊆
S

{\displaystyle S \subseteq T, T\subseteq S}

，我们就说这两个集合相等，记作

# S
T

{\displaystyle S = T}

，反之称作不相等，记作

S
≠
T

{\displaystyle S \neq T}

，相等的集合具有相同的元素。
[例句]

proper subset

**真子集**
如果

S
⊆
T

{\displaystyle S \subseteq T}

但

S
≠
T

{\displaystyle S \neq T}

，我们称

S

{\displaystyle S}

是

T

{\displaystyle T}

的**真子集**
[例句]

power set

**幂集**
由集合

S

{\displaystyle S}

的所有子集（含空集）所构成的集合，叫作

S

{\displaystyle S}

的**幂集**（power set），记作

2
S

{\displaystyle 2^S}

，如

# S
{
0
,
1
,
2
}

{\displaystyle S = \{ 0,1,2 \}}

的幂集是

2
S

=
{
∅
,
{
0
}
,
{
1
}
,
{
2
}
,
{
0
,
1
}
,
{
0
,
2
}
,
{
1
,
2
}
,
{
0
,
1
,
2
}
}
.

{\displaystyle 2^S = \{ \varnothing, \{0\}, \{1\}, \{2\}, \{0,1\}, \{0,2\}, \{1,2\}, \{0,1,2\} \}.}

[例句]

词源
--

例句
--

参见
--

Template:Navbox/11014
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欢迎来到数学词典 Wiki，本社区于2022年6月3日成立，是 中文数学 Wiki的姊妹项目。

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目前处于基础阶段，所有内容页面：

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Adrian Cook

**Adrian Cook** is a character who appears in the film *Mechanic: Resurrection* who is portrayed by Toby Eddington. He is a Sydney-based billionaire and former trafficker of underage sex workers who is Bishop's second assassination target. Bishop bypasses the tight security of Cook's penthouse apartment and breaks the glass bottom of his overhanging pool, sending him plummeting to his death.
Characters

This is a list of **characters** in the *Mechanic* franchise:

## The Mechanic (1972)

* Charles Bronson as Arthur Bishop
* Jan-Michael Vincent as Steve McKenna
* Keenan Wynn as Harry McKenna
* Jill Ireland as The Girl
* Linda Ridgeway as Louise
* Frank de Kova as The Man
* Takayuki Kubota as Yamoto
* Patrick O'Moore as The Old Man
* Enzo Fiermonte as The Mark
* Celeste Yarnall as The Mark's Girl

## The Mechanic (2011)

* Jason Statham as Arthur Bishop
* Ben Foster as Steve McKenna
* Tony Goldwyn as Dean Sanderson
* Donald Sutherland as Harry McKenna
* Jeff Chase as Burke
* John McConnell as Andrew Vaughn
* Mini Andén as Sarah
* Stuart Greer as Ralph
* Christa Campbell as Kelly
* Lance E. Nichols as Henry
* J.D. Evermore as Gun Runner
* David Leitch as Sebastian
* Mark Nutter as John Finch, Dean's Bodyguard
* Lara Grice as Mrs. Finch
* Ada Michelle Loridans as Finch's Daughter
* Katarzyna Wolejnio as Maria
* James Logan as Jorge Lara
* Eddie J. Fernandez as Lara's Guard
* Joshua Bridgewater as Raymond, The Carjacker
* Choop as Arthur The Dog

## Mechanic: Resurrection

* Jason Statham as Arthur Bishop
* Jessica Alba as Gina Thornton
* Tommy Lee Jones as Max Adams
* Michelle Yeoh as Mei
* Sam Hazeldine as Riah Crain
* John Cenatiempo as Jeremy Cocs
* Toby Eddington as Adrian Cook
* Femi Elufowoju Jr. as Marlon Krill
* Francis Tonkala Tamouya as Nuujib "The Slender Psycho"
* Rhatha Phongam as Renee Tran, Crain's Courier
* Anteo Quintavalle as Frank, Crain's Thug
* Aaron Brumfield as Adams' Head of Security
* Vithaya Pansringarm as Prison Warden
* Soji Ikai as Skinny Prisoner
* Natalie Burn as BBC Reporter
* Raicho Vasilev as Fletcher, Crain's Guard (uncredited)
The Mechanic (1972)

***The Mechanic*** is a 1972 action thriller movie directed by Michael Winner. It revolves around the story of Charles Bronson's character, Arthur Bishop, a skilled hitman who takes an inexperienced young man under his wing. The movie also stars Jan-Michael Vincent, Keenan Wynn, and Jill Ireland, and marks the first collaboration between Bronson and Winner.

## Plot

"The Mechanic" (1972) follows the story of a hitman named Arthur Bishop, who takes on an apprentice after his mentor is murdered. The two men develop a close bond, but when Bishop is forced to assassinate his friend's father, their relationship becomes strained. The film ends with a dramatic confrontation between the two hitmen.

## Cast

* Charles Bronson as Arthur Bishop
* Jan-Michael Vincent as Steve McKenna
* Keenan Wynn as Harry McKenna
* Jill Ireland as The Girl
* Linda Ridgeway as Louise
* Frank de Kova as The Man
* Takayuki Kubota as Yamoto
* Patrick O'Moore as The Old Man
* Enzo Fiermonte as The Mark
* Celeste Yarnall as The Mark's Girl
2020 Republican Party presidential primaries

**Nominee before election**Donald Trump

Presidential primaries and caucuses are to be organized by the Republican Party to select the 2,562 pledged delegates to the 2020 Republican National Convention to be held on August 17–20 to determine the party's nominee for president in the 2020 United States presidential election. The elections will take place in all 50 U.S. states, the District of Columbia, and five U.S. territories, and will occur between February 3 and August 11.

President Donald Trump announced that, due to medical reasons, he would be unable to run for a second term, thus making him the first President to voluntarily refuse to run for a consecutive second term since Lyndon B. Johnson (1963-1969) and the first from the Republicans since Calvin Coolidge (1923-1929). A total of 2 major candidates have declared their candidacies for the primaries so far.

## Candidates

Major candidates in the 2020 Republican presidential primaries are those who hold or have held significant elective office or are receiving substantial media coverage.

(age 48)
Calgary, Alberta, Canada
TBC
 Greg Abbott |  | November 13, 1957 (age 52)
Wichita Falls, Texas
TBC
Riah Crain

**Riah Crain** is a character who appears in the film *Mechanic: Resurrection* who is portrayed by Sam Hazeldine. They are wealthy and powerful arms dealer who hires Bishop to carry out three assassinations. Crain is cunning and ruthless, willing to go to great lengths to achieve their goals, including threatening to harm innocent people. They kidnap Thornton to ensure Bishop completes the assassinations.
The Mechanic Wiki

**Welcome to the The Mechanic Wiki!**
This is a wiki about the film series *Mechanic*.

We're a collaborative community website about The Mechanic that anyone, including you, can build and expand. Wikis like this one depend on readers getting involved and adding content. Click the "ADD NEW PAGE" or "EDIT" button at the top of any page to get started!

## Important articles

PlaceholderA page about your topicPlaceholderA Main CharacterPlaceholderThe First EpisodePlaceholderAn Important LocationPlaceholderA Key EventPlaceholderA Crucial Item

*Need help building out this community?*

* Rules of this wiki
* Getting Started
* How to Contribute
* Managing your new community
* Guides
* All Help articles

You can also be part of the larger Fandom family of communities. Visit Fandom's Community Central!

*Community Founders*: Write a good and paragraph-length description for your welcome section about your topic. Let your readers know what your topic is about and add some general information about it. Then you should visit the admin dashboard for more tips.
Marlon Krill

**Marlon Krill** is a character who appears in the film *Mechanic: Resurrection* who is portrayed by Femi Elufowoju Jr. He is a warlord held in a Malaysian prison who is the first of Bishop's assassination targets. Bishop gets himself imprisoned to gain Krill's trust and kills him by overdosing him with snake venom.
Mechanic: Resurrection

***Mechanic: Resurrection*** is the 2016 sequel to *The Mechanic (2011)*. It is directed by Dennis Gansel and features Jason Statham reprising his role as Arthur Bishop. This time, Bishop is forced out of retirement when his lover is kidnapped, and he must complete three difficult assassinations to save her. Tommy Lee Jones, Jessica Alba, and Michelle Yeoh join the cast in supporting roles. The movie, which was written by Philip Shelby and Tony Mosher, with a story by Shelby and Brian Pittman, is a continuation of Bishop's story from the 2011 remake.

## Plot

"Mechanic: Resurrection" (2016) sees Arthur Bishop living a quiet life until he is forced out of retirement to complete three seemingly impossible assassinations. Along the way, Bishop falls in love with a woman connected to his targets and must outwit his enemies to keep her safe. The film features high-stakes action and tense suspense as Bishop battles to complete his mission and protect the ones he loves.

## Cast

* Jason Statham as Arthur Bishop
* Jessica Alba as Gina Thornton
* Tommy Lee Jones as Max Adams
* Michelle Yeoh as Mei
* Sam Hazeldine as Riah Crain
* John Cenatiempo as Jeremy Cocs
* Toby Eddington as Adrian Cook
* Femi Elufowoju Jr. as Marlon Krill
* Francis Tonkala Tamouya as Nuujib "The Slender Psycho"
* Rhatha Phongam as Renee Tran, Crain's Courier
* Anteo Quintavalle as Frank, Crain's Thug
* Aaron Brumfield as Adams' Head of Security
* Vithaya Pansringarm as Prison Warden
* Soji Ikai as Skinny Prisoner
* Natalie Burn as BBC Reporter
* Raicho Vasilev as Fletcher, Crain's Guard (uncredited)
The Mechanic (2011)

***The Mechanic*** is a 2011 remake of the eponymous 1972 film directed by Simon West. The film stars Jason Statham in the lead role as Arthur Bishop, a methodical assassin who makes his hits look like accidents. Ben Foster plays Bishop's protégé, and Tony Goldwyn, Donald Sutherland, James Logan, Mini Andén, Jeff Chase, and Christa Campbell make up the rest of the cast. The movie was written by Lewis John Carlino and Richard Wenk.

## Plot

"The Mechanic" (2011) is a remake of the original film, with Jason Statham playing Arthur Bishop. In this version, Bishop takes on the son of one of his victims as an apprentice and begins to develop a close relationship with him. However, when his mentorship takes a dark turn, Bishop must confront his own morality and the consequences of his actions.

## Cast

* Jason Statham as Arthur Bishop
* Ben Foster as Steve McKenna
* Tony Goldwyn as Dean Sanderson
* Donald Sutherland as Harry McKenna
* Jeff Chase as Burke
* John McConnell as Andrew Vaughn
* Mini Andén as Sarah
* Stuart Greer as Ralph
* Christa Campbell as Kelly
* Lance E. Nichols as Henry
* J.D. Evermore as Gun Runner
* David Leitch as Sebastian
* Mark Nutter as John Finch, Dean's Bodyguard
* Lara Grice as Mrs. Finch
* Ada Michelle Loridans as Finch's Daughter
* Katarzyna Wolejnio as Maria
* James Logan as Jorge Lara
* Eddie J. Fernandez as Lara's Guard
* Joshua Bridgewater as Raymond, The Carjacker
* Choop as Arthur The Dog
Mei

**Mei** is a character who appears in the film *Mechanic: Resurrection* who is portrayed by Michelle Yeoh. She is a friend of Bishop who lives on a resort island beach house in Thailand. Bishop seeks refuge at Mei's house after escaping from Renee Tran. Mei helps Bishop in his quest to stop Crain's evil plan.
Set operation

**集合运算**基本术语主题。

术语
--

union set

**并集**

A
∪
# B
{
x
∈

Ω

x
∈
A
,
or
x
∈
B
}
;

{\displaystyle A \cup B = \{ x \in \mathit\Omega | x \in A, \text{or } x \in B \};}

[例句]

intersection set

**交集**

A
∩
# B
{
x
∈

Ω

x
∈
A
,
and
x
∈
B
}
;

{\displaystyle A \cap B = \{ x \in \mathit\Omega | x \in A, \text{and } x \in B \};}

[例句]

disjoint union

**不交并**
当

A
∩
# B
∅

{\displaystyle A \cap B = \varnothing}

时称

A
∪
B

{\displaystyle A \cup B}

为这两个集合的不交并。
[例句]

difference set

**差集**

A
∖
# B
A
−
# B
{
x
∈

Ω

x
∈
A
,
and
x
∉
B
}
.

{\displaystyle A \setminus B = A - B = \{ x \in \mathit\Omega | x \in A, \text{and } x \notin B \}.}

[例句]

Cartesian product set

**集合的笛卡尔积**

A
×
# B
{
(
x
,
y
)

x
∈
A
,
y
∈
B
}
.

{\displaystyle A \times B = \{ (x, y) | x \in A, y \in B \}.}

[例句]

complement of a subset

**子集的补集**

A
C

=

A
¯

:=
{
x
∈

Ω

x
∉
A
}
.

{\displaystyle A^C = \overline{A} := \{ x \in \mathit\Omega | x \notin A\}.}

[例句]

词源
--

例句
--

参见
--

Template:Navbox/11014
Renee Tran

**Renee Tran** is a character who appears in the film *Mechanic: Resurrection* who is portrayed by Rhatha Phongam. He is a courier who approaches Bishop in Rio de Janeiro, knows his true identity, and tries to hire him to carry out three assassinations. When Bishop refuses, Tran sends her mercenaries after him.
Define

**Define** 是动词，定义。在数学中经常使用以定义新概念。

词汇
--

define
vt.
定义，下定义
[例句]

definition
nc.
定义
[例句]

well-defined
adj.
**良定义的**
主要说明这种定义是合理的，采用公理化定义时各个公理之间不会冲突、一般的定义出来有元素满足（即不是空集）、定义不会出现歧义，例如，定义一个满足某些条件的映射时需要注意不能一对多。
[例句]

ill-defined
adj.
**劣定义的**
可以指定义有歧义的，模棱两可的，也可以指定义不规范的
[例句]

词源
--

例句
--

1. It is often better to adopt *definition*s that express the elements of a set as elements s of some larger (and already known) set S, satisfying some property P.
最好采用将一个集合的元素表达为更大的（已知）集合中满足某些性质的元素这样的定义方式。

参见
--
Number set

**数集**基本术语主题。

术语
--

number set

数集，由数字构成的集合。
[例句]

natural number set

**自然数集，可以有零，也可以没有，根据每个人不同的定义方式。**

N

=
{
1
,
2
,
3
,
⋯
}
;

{\displaystyle \N^+ = \{1,2,3,\cdots \};}

[例句]

nonnegative integer set

**非负整数集，有零**

{
0
,
1
,
2
,
3
,
⋯
}
;

{\displaystyle \{0,1,2,3,\cdots \};}

[例句]

integer set

**整数集**

Z

=
{
0
,
±
1
,
±
2
,
⋯
}
;

{\displaystyle \Z = \{0,\pm1,\pm2,\cdots\};}

[例句]

rational number set

**有理数集**

Q

=
{

p
q

(
p
,
q
# )
1
,
p
,
q
∈

N

,
q
>
0
}
;

{\displaystyle \Q = \{\dfrac{p}{q} | (p,q) = 1, p,q \in \N, q>0\};}

[例句]

real number set

实数集，

R

;

{\displaystyle \R;}

[例句]

complex number set

**复数集**

C

=
{
a
i
b

a
,
b
∈

R

,

i
2

=
−
1
}
.

{\displaystyle \C = \{a+\text{i}b|a,b\in\R,\text{i}^2 = -1\}.}

[例句]

词源
--

例句
--

参见
--

Template:Navbox/11014
PCAFI

The PCAFI or the Pentecostal Churches of the Apostolic Faith International, more commonly referred to as the PCAF, is one of the main parts of the Pentecostal Faith. It is mainly compared to The PAW

The PAW is currently shepherded by Bishop Lambert W. Gates, Sr. He is assisted by both Bishop Mark C. Tolbert and Bishop Michael D. Hannah, Sr.

https://www.pcafintl.org/copy-of-history
Identity

Identity 是单位元、恒等元。

词汇
--

left-identity
nc.
**左单位元，左幺元**
若

∃

e
l

∈

S

,
∀
x
∈

S

,

e
l

∘
# x
x

{\displaystyle \exists e\_l \in \mathbf{S}, \forall x \in \mathbf{S}, e\_l \circ x = x}

，称

e
l

{\displaystyle e\_l}

为运算

∘

{\displaystyle \circ}

的左单位元
[例句]

right-identity
nc.
**右单位元，右幺元**
若

∃

e
r

∈

S

,
∀
x
∈

S

,
x
∘

e
r

=
x

{\displaystyle \exists e\_r \in \mathbf{S}, \forall x \in \mathbf{S}, x \circ e\_r = x}

，称

e
r

{\displaystyle e\_r}

为运算

∘

{\displaystyle \circ}

的右单位元
[例句]

identity
nc.
**单位元，幺元**
当

e
l

=

e
r

{\displaystyle e\_l = e\_r}

时，称其为运算

∘

{\displaystyle \circ}

的单位元，因此也称 two-sided identity
[例句]

inverse
nc.
逆（参见Inverse）
[例句]

left-unit
nc.
**（环的）左单位**
一个环

R

{\displaystyle R}

中的元素

u

{\displaystyle u}

称为是左单位是指存在

v
∈
R

{\displaystyle v \in R}

使得

u
# v
1

{\displaystyle uv = 1}

[例句]

right-unit
nc.
**（环的）右单位**
一个环

R

{\displaystyle R}

中的元素

u

{\displaystyle u}

称为是右单位是指存在

v
∈
R

{\displaystyle v \in R}

使得

v
# u
1

{\displaystyle vu = 1}

[例句]

unit
nc.
**（环的）单位**
一个环

R

{\displaystyle R}

中的元素

u

{\displaystyle u}

称为是单位是指它既是左单位也是右单位，即 two-sided unit
[例句]

词源
--

例句
--

1. The reader should keep in mind that inverses of left- or right-units are not unique in general, so the ‘inverse notation’ is not appropriate for them.
读者应谨记，左或右单位的逆通常不是唯一的，因此“逆符号”不适用于它们。

参见
--
Algebra

**代数**的基本术语主题。

术语
--

algebra
n.
**代数**
使用抽象符号代替具体数字，并研究符号间的关系与整体结构的数学学科。
[例句]

linear algebra

**线性代数**
从经典的欧氏几何抽象出的代数结构，也指研究这一结构的数学学科，包括矩阵，行列式，线性方程组，线性空间和二次型等理论。
[例句]

modern algebra
abstract algebra

**近世代数，抽象代数**
近代数学发展的更抽象结果，对一个集合不断施加各种运算得到的代数结构的研究，包括群论和半群论，环论和模论，域论和伽罗瓦理论等。
[例句]

Lie algebra

李代数
[例句]

universal algebra

泛代数
[例句]

homology algebra

同调代数
[例句]

词源
--

例句
--

参见
--

Template:Navbox/11014
Canonical

Canonical的意思是典范的，标准的。这个词涉及到的词汇都带有一种“自然的”“性质良好的”的意思。

词汇
--

canonical projection

**典范投射，标准投射**
一种将一个代数结构做商的映射，如群

G

{\displaystyle G}

若有正规子群

H

{\displaystyle H}

，那么

π
:
G
→
G

/

H

{\displaystyle \pi: G \to G/H}

，它将

G

{\displaystyle G}

的元素映成对应的陪集，这样的投射就是典范的
[例句]

canonical decomposition

**典范分解，标准分解**
将一个同态分解为一个满射，双射和单射的复合的过程，如群同态

f
:
G
→
H

{\displaystyle f: G \to H}

的典范分解

G

⟶
π

G

/

Ker
⁡
φ

⟶

φ
~

Im
⁡
φ

⟶
j

H
.

{\displaystyle \begin{align}
G \overset{\pi}{\longrightarrow} G/\operatorname{Ker} \varphi \overset{\tilde{\varphi}}{\longrightarrow} \operatorname{Im} \varphi \overset{j}{\longrightarrow} H.
\end{align}}

[例句]

canonical form

**标准型**
矩阵的标准型，例如等价标准型、合同标准型和相似标准型等等
[例句]

词源
--

例句
--

参见
--
Equivalence relation

**等价关系**的相关术语主题。

词汇
--

relation
nc.
关系，详见 relation
[例句]

equivalence relation

**等价关系**
设

R

{\displaystyle R}

是集合

A

{\displaystyle A}

上的一个二元关系，称其为一个等价关系，如果它满足以下条件：
1. 自反性：

∀
x
∈
A
,
x
R
x

{\displaystyle \forall x \in A, xRx}

；
2. 对称性：

∀
x
,
y
∈
A
,
x
R
y

⟹

y
R
x

{\displaystyle \forall x,y\in A,xRy\implies yRx}

；
3. 传递性：

∀
x
,
y
,
z
∈
A
,
(
x
R
y
∧
y
R
z
)

⟹

x
R
z

{\displaystyle \forall x,y,z\in A,(xRy\wedge yRz)\implies xRz}

。

等价关系的符号也可写为

∼

{\displaystyle \sim}

[例句]

be equivalent to

**等价于**
若

(
x
,
y
)
∈
R

{\displaystyle (x,y) \in R}

，称

x

{\displaystyle x}

等价于

y

{\displaystyle y}

，记作

x
∼
y

{\displaystyle x \sim y}

[例句]

equivalence class

**等价类**
设

R

{\displaystyle R}

是集合

A

{\displaystyle A}

上的一个等价关系，

A

{\displaystyle A}

中等价于元素

x

{\displaystyle x}

的所有元素组成的集合称为

x

{\displaystyle x}

的等价类，记作

[
x

]

R

=

x
¯

=
{
y
∈
A

x
R
y
}

{\displaystyle [x]\_{R}={\bar {x}}=\{y\in A|xRy\}}

[例句]

quotient set

**商集**
设

R

{\displaystyle R}

是非空集合

A

{\displaystyle A}

上的一个等价关系，以

R

{\displaystyle R}

的所有等价类作为元素的集合称为

A

{\displaystyle A}

关于

R

{\displaystyle R}

的商集（合），记作

A

/

R

{\displaystyle A/R}

[例句]

partition
n.
**分划**
集合

A

{\displaystyle A}

的子集族

π

{\displaystyle \pi }

若满足：
1. 不含空集，即

∅
⊈
π

{\displaystyle \varnothing \nsubseteq \pi}

；
2. 任意两元素不相交；
3. 所有元素的并等于

A

{\displaystyle A}

，

则称

π

{\displaystyle \pi }

为

A

{\displaystyle A}

的分划，记作

π
R

=
{

a
¯

}

a
¯

∈
R

{\displaystyle \pi\_R = \{ \bar{a} \}\_{\bar{a} \in R}}

[例句]

词源
--

例句
--

参见
--
Churches Pastors/Bishops Wiki

**Welcome to the Churches Pastors/Bishops Wiki!**
These articles describe where Churches are and what pastors reside there. It mainly focuses on Pentecostal and Apostolic Churches

 **Important articles**

*Need help building out this community?*

* Rules of this wiki
* Getting Started
* How to Contribute
* Managing your new community
* Guides
* All Help articles

You can also be part of the larger Fandom family of communities. Visit Fandom's Community Central!

*Community Founders*: Write a good and paragraph-length description for your welcome section about your topic. Let your readers know what your topic is about and add some general information about it. Then you should visit the admin dashboard for more tips.
Operation law

**运算律**的基本术语主题。

术语
--

operation law
nc.
**运算律**
指一个代数结构

S

{\displaystyle S}

中一些元素关于某运算

∗

{\displaystyle \*}

所满足的恒等式或不等式
[例句]

associative
adj.
**可结合的，满足结合律的**

a
,
b
,
c
∈
S
,
a
∗
(
b
∗
c
# )
(
a
∗
b
)
∗
c

{\displaystyle a, b, c \in S, a\*(b\*c) = (a\*b)\*c}

矩阵乘法就满足结合律
[例句]

commutative
adj.
**可交换的，满足交换律的**

a
,
b
∈
S
,
a
∗
# b
b
∗
a

{\displaystyle a, b \in S, a\*b = b\*a}

数的加法和乘法都满足交换律
[例句]

idempotent
adj.
**幂等的，满足幂等律的**

a
∈
S
,
a
∗
# a
a

{\displaystyle a \in S, a\*a = a}

[例句]

distributive
adj.
**可分配的，满足分配律的**

a
,
b
,
c
∈
S
,
a
(
b
c
# )
a
c
b
c

{\displaystyle a, b, c \in S, a(b+c) = ac + bc}

，这是左分配律，可以仿照写出右分配律
[例句]

词源
--

例句
--

参见
--
Mel

was a Worker sent by PepsiCo. to assist Saul Goodman on how to deal with Idaho and to represent their company. Though not much is known about his personal life, DougDoug seems to know him personally from his words "Classic Mel".

He was called upon by Saul Goodman along with other representatives of other companies under Dougkota. On the meeting, he was asked by Saul Goodman on how they can convince the Idahoans to join Dougkota as his other strategies did not seem to be effective. Mel then suggest that they should try Advertising, that with enough money they could sell the Idahoans corn flakes and they would in turn, join the Empire.

Saul Goodman then ask Mel how much money is needed. Mel simply said that he does not know, but he then said that if you want to be rich, you gotta spend money!. Saul Goodman then proceeded to run a huge ad campaign for months together with all the companies in Dougkota. They advertised the Empire's corn harvest, and they said that Dougkota would give them everything they need to get wealthy.

And it worked. The Idahoans believed there propositions and decided to join Dougkota. Idaho added 50,000 soldiers to Dougkota's army.
2020 United States presidential election

The **2020 United States presidential election** will be the 59th quadrennial presidential election, scheduled for Tuesday, November 3, 2020. Incumbent president Donald Trump has stated that he does not intens to run for reelection to a second term, despite his victory in 2016, due to health issues.

In the United States, general elections follow caucuses and primary elections held by the major parties to determine their nominees. The winner of the 2020 presidential election is scheduled to be inaugurated on January 20, 2021.
Integer

**Integer** 主题相关的词汇。

词汇
--

integer
nc.
整数
[例句]

even integer

偶数
[例句]

odd integer

奇数
[例句]

prime integer

素数
[例句]

composite number

合数
[例句]

multiple
nc.
倍数（也可作动词）
[例句]

common multiple

公倍数
[例句]

least common multiple (lcm)

**最小公倍数**
可简写为 lcm，函数形式为

lcm

(
a
,
b
)
,
[
a
,
b
]

{\displaystyle {\text{lcm}}(a,b),[a,b]}

[例句]

divisor, factor
nc.
因数，约数
[例句]

common divisor, common factor

公因数，公约数
[例句]

greatest common divisor, greatest common factor

**最大公因数，最大公约数**
可简写为 gcd 或 hcf，函数形式为

gcd
(
a
,
b
)
,
(
a
,
b
)

{\displaystyle \gcd(a,b),(a,b)}

[例句]

relatively prime to n

**与

n

{\displaystyle n}

互素**
如果

m
,
n

{\displaystyle m,n}

互素，一般表示为

(
m
,
n
# )
1

{\displaystyle (m,n)=1}

[例句]

词源
--

例句
--

参见
--
United States Roleplay Season 1 Wiki

**Welcome to the United States Roleplay Season 1 Wiki!**
community

We're a collaborative community website about United States Roleplay Season 1 that anyone, including you, can build and expand. Wikis like this one depend on readers getting involved and adding content. Click the "ADD NEW PAGE" or "EDIT" button at the top of any page to get started!

## Important articles

Marsha CooperPlaceholderA Main CharacterPlaceholderThe First EpisodePlaceholderAn Important LocationPlaceholderA Key EventPlaceholderA Crucial Item

*Need help building out this community?*

* Rules of this wiki
* Getting Started
* How to Contribute
* Managing your new community
* Guides
* All Help articles

You can also be part of the larger Fandom family of communities. Visit Fandom's Community Central!

*Community Founders*: Write a good and paragraph-length description for your welcome section about your topic. Let your readers know what your topic is about and add some general information about it. Then you should visit the admin dashboard for more tips.
Finite

**Finite**相关词汇列表。

词汇
--

finite
adj.
**有限的**
一个代数结构（如集合，群）有限是说其中的元素是有限个
[例句]

finite number

**有限数**
正常意义的数，和无穷做对比
[例句]

infinite
adj.
**无限的**
一个代数结构（如集合，群）有限是说其中的元素是无限的
[例句]

infinite number

**无穷大量**
扩展实数上是指正负无穷大

±
∞

{\displaystyle \pm\infty}

（infty），严格来说它并不是一般意义的数字，而是极限过程，复数上的无穷大是

∞

{\displaystyle \infty}

[例句]

finitely generated

**有限生成的**
代数系统中的概念，一个代数系统是有限生成的是说其中的元素可以被有限个元素通过这个系统中的运算生成，这些生成其它元素的元素就称为生成元。
[例句]

词源
--

例句
--

参见
--
Gina Thornton

**Gina Thornton** is a character who appears in the film *Mechanic: Resurrection* who is portrayed by Jessica Alba. She is a woman Bishop rescues from a man who is beating her on a boat anchored near Mei's beach house. Thornton is connected to Crain and participates in her plan to force Bishop to carry out the assassinations by threatening the children's shelter in Cambodia that she runs. Thornton falls in love with Bishop, and together, they work to stop Crain's plan.
Product

**积**的相关术语主题。

词汇
--

product
nc.
**积**
从一些代数系统（如群，环，模等）或范畴衍生出新的同类型代数系统的方式。
[例句]

Cartesian product

**笛卡尔积**
集合定义的一种自然乘积，

S
,
T

{\displaystyle S, T}

的笛卡尔积是指

S
×
T
:=
{
(
s
,
t
)

s
∈
S
,
t
∈
T
}

{\displaystyle S \times T := \{ (s, t) | s \in S, t \in T \}}

[例句]

direct product
nc.
**直积**
积的一种，以“自然的”坐标对组合的方式形成新的代数系统，例如设有两个群

G
,
H

{\displaystyle G, H}

，我们在集合

G

{\displaystyle G}

和

H

{\displaystyle H}

的笛卡尔积

G
×
# H
{
(
g
,
h
)

g
∈
G
,
h
∈
H
}

{\displaystyle G \times H = \{ (g, h) | g \in G, h \in H \}}

上定义如下乘法运算：

(

g
1

,

h
1

)
⋅
(

g
2

,

h
2

# )
(

g
1

⋅

g
2

,

h
1

⋅

h
2

)
.

{\displaystyle (g\_1, h\_1) \cdot (g\_2, h\_2) = (g\_1 \cdot g\_2, h\_1 \cdot h\_2).}

可以验证

(
G
×
H
,
⋅
)

{\displaystyle (G \times H, \cdot)}

构成一群，我们称其为

G

{\displaystyle G}

和

H

{\displaystyle H}

的直积
[例句]

semidirect product

**半直积**
详见 w:c:zh.math:半直积
[例句]

coproduct
nc.
**余积**
范畴上积的余概念
[例句]

direct sum

**直和**
交换群范畴以及模范畴上的概念，可交换的代数结构上积和余积是一样的情形下统称它们为直和
[例句]

fibered product

**纤维积**
范畴上的一种积
[例句]

componentwise
adj.
按分量定义的
[例句]

free product

**自由积**
余积的另外一个称呼，因为余积可以通过自由对象定义，故名
[例句]

词源
--

例句
--

1. With operation defined componentwise,

G
×
H

{\displaystyle G \times H}

 is a product in

G
r
p

{\displaystyle \mathsf{Grp}}

.
在按分量定义（乘法）运算后，

G
×
H

{\displaystyle G \times H}

是群范畴

G
r
p

{\displaystyle \mathsf{Grp}}

中的积。
2. The coproduct of

G

{\displaystyle G}

 and

H

{\displaystyle H}

 in

G
r
p

{\displaystyle \mathsf{Grp}}

 is denoted

G
∗
H

{\displaystyle G \* H}

 and is called the free product of

G

{\displaystyle G}

 and

H

{\displaystyle H}

.

G
r
p

{\displaystyle \mathsf{Grp}}

中

G

{\displaystyle G}

和

H

{\displaystyle H}

的余积被记作

G
∗
H

{\displaystyle G \* H}

，同时也称作

G

{\displaystyle G}

和

H

{\displaystyle H}

的自由积。
3. When working as a coproduct, the product

G
×
H

{\displaystyle G \times H}

 of two abelian groups is often called their direct sum.
两个交换群的积

G
×
H

{\displaystyle G \times H}

同时也作为余积时，它就被称为

G

{\displaystyle G}

和

H

{\displaystyle H}

的直和。

参见
--
Saul Goodman

**Saul Goodman** (born November 12, 1960) is an American lawyer and politician who has served as the junior Senator from New Mexico in the United States Senate since 2013 as a member of the Democratic Party. He had previously served as a city councillor in Albuquerque, and as Attorney General of New Mexico.

Born in Cicero, Illinois, Saul moved to Albuquerque to pursue a career as a criminal lawyer, eventually graduating from the University of New Mexico and taking up a role as a lawyer at the firm Hamlin, Hamlin & McGill. It was here that Saul met Kim Wexler, a fellow lawyer, whom he later married. Saul served on Albuquerque City Council for District 6 from 2003 to 2007, before taking on the post of Attorney General of New Mexico. Saul won election to the United States Senate for New Mexico in 2012, defeating Republican Heather Wilson, and was sworn in on January 3rd 2013. Saul has been a long-time advocate of progressive, social democratic policies since he entered politics, his time as a criminal lawyer having exposed him to a lot of the socio-economic inequality rampant in Albuquerque and America as a whole.

On November 1st 2019, Saul declared his candidacy in the 2020 Democratic Party presidential primaries. He quickly secured several major endorsements, such as his colleague in the Senate Marsha Cooper and Speaker of the House of Representatives Nancy Pelosi. He held his first Presidential campaign rally on the same day in Albuquerque, expressing support for policies such as Medicare For All and a Green New Deal, and condemning the Donald Trump administration.
Homomorphism

**同态**相关术语主题。

词汇
--

homomorphism
nc.
**同态**
描述两个代数结构之间关系的集函数（映射），保持了代数结构间的运算以及某些特殊元素（例如单位元，零元等）
* group ~群同态
* ring ~环同态
* module ~模同态

[例句]

monomorphism
nc.
**单态射**
范畴上的概念，一个代数系统中的同态（态射）如果左可消，即设

A
,
B
,
C
∈
Obj
⁡

C

,
f
:
A
→
B

{\displaystyle A,B,C\in \operatorname {Obj} {\mathsf {C}},f:A\to B}

，若

∀
g
,
h
:
C
→
A

{\displaystyle \forall g,h:C\to A}

使得

f
∘
# g
f
∘
h
⟹
# g
h
.

{\displaystyle f\circ g=f\circ h\Longrightarrow g=h.}

那么

f

{\displaystyle f}

就是单态射。
[例句]

epimorphism
nc.
**满态射**
范畴上的概念，一个代数系统中的同态（态射）如果右可消，即设

A
,
B
,
C
∈
Obj
⁡

C

,
f
:
A
→
B

{\displaystyle A,B,C\in \operatorname {Obj} {\mathsf {C}},f:A\to B}

，若

∀
g
,
h
:
C
→
A

{\displaystyle \forall g,h:C\to A}

使得

g
∘
# f
h
∘
f
⟹
# g
h
.

{\displaystyle g\circ f=h\circ f\Longrightarrow g=h.}

那么

f

{\displaystyle f}

就是满态射。
[例句]

isomorphism
nc.
**同构**
代数系统中的同构是指一个映射作为集合函数既是单射又是满射，范畴中的对象同构是：在范畴

C

{\displaystyle {\mathsf {C}}}

中，设

A
,
B
∈
Obj
⁡

C

,
f
:
A
→
B

{\displaystyle A,B\in \operatorname {Obj} {\mathsf {C}},f:A\to B}

，若存在

g
:
B
→
A

{\displaystyle g:B\to A}

使得下式成立

g
∘
# f

1

A

,

f
∘
# g

1

B

.

{\displaystyle g\circ f=1\_{A},\quad f\circ g=1\_{B}.}

我们就称

f

{\displaystyle f}

是一个同构
[例句]

bimorphism
nc.
**双态射**
既是单态射又是满态射的态射，称为双态射。同构是双态射，反之不然。
[例句]

image
nc.
**同态象**
设

f
:
G
→
H

{\displaystyle f:G\to H}

是同态，

Im
⁡
φ
:=
{
φ
(
a
)

a
∈
G
}
=:
φ
(
G
)

{\displaystyle \operatorname {Im} \varphi :=\{\varphi (a)|a\in G\}=:\varphi (G)}

称作

φ

{\displaystyle \varphi }

的象，在环和模（特例，线性空间）以及域中有类似概念
[例句]

kernel
nc.
**同态核**
设

f
:
G
→
H

{\displaystyle f:G\to H}

是同态，

Ker
⁡
φ
:=
{
a
∈
G

φ
(
a
# )

e

G
′

}
=:

φ

−
1

(

G
′

)

{\displaystyle \operatorname {Ker} \varphi :=\{a\in G|\varphi (a)=e\_{G'}\}=:\varphi ^{-1}(G')}

称作

φ

{\displaystyle \varphi }

的核，在环和模（特例，线性空间）以及域中有类似概念
[例句]

coimage
nc.
**同态的余象**
范畴中同态象的余概念，在模范畴中良定义
[例句]

cokernel
nc.
**同态的余核**
范畴中同态核的余概念，在交换群范畴和模范畴中良定义
[例句]

词源
--

例句
--

参见
--
Ring

**环**的相关术语主题。

词汇
--

ring
nc.
**环**
我们称一个代数系统

(
R
,
,
⋅
)

{\displaystyle (R, +, \cdot)}

是一个环，如果满足：
1. (
R
,
)

{\displaystyle (R, +)}

是一个交换群；
2. (
R
,
⋅
)

{\displaystyle (R, \cdot)}

是一个幺半群；
3. 乘法

⋅

{\displaystyle \cdot}

对加法

{\displaystyle +}

满足分配律

∀
a
,
b
,
c
∈
R
,

a
(
b
c
# )
a
b
a
c
,

(
a
b
)
# c
a
c
b
c
.

{\displaystyle \forall a, b, c \in R, \quad a(b+c) = ab + ac, \quad (a+b)c = ac + bc.}

一般我们也将

(
R
,
,
⋅
)

{\displaystyle (R, +, \cdot)}

简记作

R

{\displaystyle R}

[例句]

rng
nc.
**无幺环，有时也指环范畴**
在环的定义中将“

(
R
,
⋅
)

{\displaystyle (R, \cdot)}

是一个幺半群”改为“

(
R
,
⋅
)

{\displaystyle (R, \cdot)}

是一个半群”
[例句]

zero-ring
nc.
**零环**
只有一个元素的环，这个元素同时是零元和单位元，记作

{
0
}

{\displaystyle \{0\}}

[例句]

commutative ring,
Abelian ring

**交换环，阿贝尔环**
环中元素对乘法满足交换律的环
[例句]

left-zero-divisor,
right-zero-divisor,
zero-divisor
nc.
**左零因子，右零因子，零因子**
设

R

{\displaystyle R}

是环，如果存在

a
∈
R

{\displaystyle a \in R}

，对任意的

b
∈
R
,
b
≠
a

{\displaystyle b \in R, b \ne a}

都有

a
# b
0

{\displaystyle ab = 0}

我们就说

a

{\displaystyle a}

是一个左零因子，同样可以定义右零因子

a

{\displaystyle a}

，它满足的条件是

b
# a
0

{\displaystyle ba = 0}

如果

a

{\displaystyle a}

既是左零因子又是右零因子，我们就称

a

{\displaystyle a}

是零因子
[例句]

integral domain
nc.
**整环**
无零因子的交换环，

∀
a
,
b
∈
R
,
a
# b
0
⟹
# a
0
or
# b
0

{\displaystyle \forall a, b \in R, ab = 0 \Longrightarrow a = 0 \text{or} b = 0}

[例句]

non-zero-divisor
nc.
非零因子
[例句]

unique factorization domain

唯一分解整环, UFD
[例句]

principal ideal domain

主理想整环, PID
[例句]

Euclidean domain

欧几里得（整）环
[例句]

unit

单位，详见 Identity
[例句]

division ring

**除环，体**
每一个非零因子都是单位的环
[例句]

polynomial ring

多项式环，详见 w:c:zh.math:多项式环
[例句]

ring homomorphism

**环同态，详见 Homomorphism**
设有两个环

R
,
S

{\displaystyle R, S}

，如果一映射

φ
:
R
→
S

{\displaystyle \varphi: R \to S}

满足
1. 保持加法：

∀
a
,
b
∈
R
,
φ
(
a
b
# )
φ
(
a
)
φ
(
b
)
;

{\displaystyle \forall a, b \in R, \varphi(a + b) = \varphi(a) + \varphi(b);}
2. 保持乘法：

∀
a
,
b
∈
R
,
φ
(
a
⋅
b
# )
φ
(
a
)
⋅
φ
(
b
)
;

{\displaystyle \forall a, b \in R, \varphi(a \cdot b) = \varphi(a) \cdot \varphi(b);}
3. 保持单位元：

φ
(

1
R

# )

1
S

.

{\displaystyle \varphi(1\_R) = 1\_S.}

我们就说

φ

{\displaystyle \varphi}

是

R

{\displaystyle R}

到

S

{\displaystyle S}

的一个环同态
[例句]

subring
nc.
**子环**
设有一环

R

{\displaystyle R}

，设有

R

{\displaystyle R}

的非空子集

S

{\displaystyle S}

，如果

S

{\displaystyle S}

到

R

{\displaystyle R}

存在一嵌入

f

{\displaystyle f}

是环同态（即

f

{\displaystyle f}

是一单同态），我们就称

S

{\displaystyle S}

是

R

{\displaystyle R}

的子环
[例句]

ideal
nc.
理想，详见 ideal
[例句]

quotient ring

商环，详见 w:c:zh.math:商环
[例句]

Boolean ring

布尔环
[例句]

characteristic
nc.
**（环的）特征**
设有环

R

{\displaystyle R}

以及环同态

f
:

Z

→
R
,
n
↦
n
⋅

1
R

{\displaystyle f: \mathbb{Z} \to R, n \mapsto n \cdot 1\_R}

，那么

ker
⁡
f

{\displaystyle \ker f}

中最小的正整数就称为环

R

{\displaystyle R}

的特征，如果不存在这样的正整数，那么该环的特征为正无穷大
[例句]

词源
--

例句
--

参见
--
Relation

**集合上的关系**的相关术语主题。

词汇
--

relation on a set

集合上的关系
[例句]

binary relation

**二元关系**
给定一集合

A

{\displaystyle A}

，如果一个集合

R

{\displaystyle R}

满足以下条件之一：
1. 为空集（

∅

{\displaystyle \varnothing}

）
2. 非空集合且元素都是二元有序对

(
x
,
y
)
,
x
,
y
∈
A

{\displaystyle (x, y), x,y \in A}

则称该集合为一个二元关系（或简称关系）。对于

R

{\displaystyle R}

，如果

(
x
,
y
)
∈
R

{\displaystyle (x,y) \in R}

 ，则记作

x
R
y

{\displaystyle xRy}

，意思就是

x

{\displaystyle x}

和

y

{\displaystyle y}

有

R

{\displaystyle R}

关系，若

(
x
,
y
)
∉
R

{\displaystyle (x,y) \notin R}

，则记作

x

⧸

R
⁡
y

{\displaystyle x \not\operatorname{R} y}

[例句]

reflexivity
n.
**自反性**
形容词形式为 reflexive。称

R

{\displaystyle R}

是自反的，如果

∀
x
∈
A
,
(
x
,
x
)
∈
R

{\displaystyle \forall x \in A, (x, x) \in R}

[例句]

irreflexivity
n.
**反自反性**
形容词形式为 irreflexive。称

R

{\displaystyle R}

是反自反的，如果

∀
x
∈
A
,
(
x
,
x
)
∉
R

{\displaystyle \forall x \in A, (x,x) \notin R}

[例句]

symmetry
n.
**对称性**
形容词形式为 symmetric。称

R

{\displaystyle R}

是对称的，如果

∀
x
,
y
∈
A
,
x
R
y
⇒
y
R
x

{\displaystyle \forall x, y \in A, xRy \Rightarrow yRx}

[例句]

antisymmetry
n.
**反对称性**
形容词形式为 antisymmetric。称

R

{\displaystyle R}

是反对称的，如果

∀
x
,
y
∈
A
,
(
x
R
y
∧
y
R
x
)
⇒
# x
y

{\displaystyle \forall x, y \in A, ( xRy \land yRx ) \Rightarrow x = y}

[例句]

transitivity
n.
**传递性**
形容词形式为 transitive。称

R

{\displaystyle R}

是传递的，如果

∀
x
,
y
,
z
∈
A
,
(
x
R
y
∧
y
R
z
)
⇒
x
R
z

{\displaystyle \forall x, y, z \in A, ( xRy \land yRz) \Rightarrow xRz}

[例句]

equivalence relation

等价关系，详见 equivalence relation
[例句]

partial order relation

偏序关系，详见 order relation
[例句]

词源
--

例句
--

参见
--
Set

**集合**的基本术语主题。

术语
--

set
nc.
**集合**
指包含「一堆东西」的「东西」，一个集合里的东西称为「元素」。集合一般使用大写英文字字母表示，集合的表示方法有两种：一种是直接列出所有元素，并用大括号包围起来，如

# S
{
1
,
2
,
3
}

{\displaystyle S = \{ 1,2,3 \}}

表示由数字1，2，3组成的一个集合，

# S
{
(
1
,
1
)
,
(
0
,
2
)
}

{\displaystyle S = \{ (1,1), (0,2) \}}

表示由点

(
1
,
1
)
,
(
0
,
2
)

{\displaystyle (1,1), (0,2)}

组成的一个集合。另一种表示是规定元素具有的某种性质所表示的集合
[例句]

element
nc.
**元素**
集合中的对象。一个元素

a

{\displaystyle a}

如果在集合

S

{\displaystyle S}

中，记作

a
∈
S

{\displaystyle a \in S}

，如果不在集合

S

{\displaystyle S}

中记作

a
∉
S

{\displaystyle a \notin S}

或

a

∈
¯

S
.

{\displaystyle a \overline{\in} S.}

[例句]

multiset
nc.
**多重集**
集合定义中不能有重复的元素出现，而多重集则允许出现重复的元素，例如

# S
{
1
,
1
,
1
,
2
,
2
,
3
}

{\displaystyle S = \{ 1,1,1,2,2,3 \}}

就是一个具有六个元素的多重集，两个多重集相等必须有相同的元素，且相同的元素出现的次数一样。
[例句]

empty set

空集，一般用记号

∅

{\displaystyle \varnothing}

表示
[例句]

singleton
nc.
**单点集**
只有一个元素的集合，如

{
1
}
,
{
A
}

{\displaystyle \{ 1 \}, \{ A \}}

，源于词 single.
[例句]

coset
nc.
**陪集**
代数系统的概念，一个群

G

{\displaystyle G}

有子群

H

{\displaystyle H}

，那么

H

{\displaystyle H}

关于

a

{\displaystyle a}

的（左）陪集

a
# H
{
h
∈
G

∃
x
∈
H
,
# h
a
x
}

{\displaystyle aH = \{ h \in G | \exists x \in H, h = a x \}}

，环中也有类似的概念：一个环

R

{\displaystyle R}

有子环

I

{\displaystyle I}

，那么

I

{\displaystyle I}

关于

r
∈
R

{\displaystyle r \in R}

的（左）陪集

r
# I
{
i
∈
R

∃
x
∈
I
,
# i
r
x
}

{\displaystyle r + I = \{ i \in R | \exists x \in I, i = r + x \}}

[例句]

indexed set
nc.
指标集
[例句]

词源
--

例句
--

1. A *multiset* is a set in which the elements are allowed to appear ‘with multiplicity’: that is, a notion for which

{
2
,
2
}

{\displaystyle \{2,2\}}

 would be distinct from

{
2
}

{\displaystyle \{2\}}

.
一个多重集是一种允许元素多次出现的集合：也就是说，

{
2
,
2
}

{\displaystyle \{2,2\}}

表示的概念会和

{
2
}

{\displaystyle \{2\}}

不同。
2. The *empty set* is the set containing no elements.
空集是没有元素的集合。
3. Also, the term *singleton* is used to refer to any set consisting of precisely one element. Thus

{
1
}
,
{
2
}
,
{
3
}

{\displaystyle \{1\}, \{2\}, \{3\}}

 are different sets, but they are all singletons.
同时，单点集这个词汇被用来指仅含有一个元素的任意一个集合。因此

{
1
}
,
{
2
}
,
{
3
}

{\displaystyle \{1\}, \{2\}, \{3\}}

虽是不同的集合但它们都是单点集。

参见
--

Template:Navbox/11014
Tami

## Locations

* Downtown - Weekend Mornings
* Suburbs - Weekend Evenings

## Entries

I now know where Tora gets her flirtatious attitude from. Tami Chura… I’m not sure if I like her or if I’m scared of her… I think it’s a little bit of both.  I haven’t seen Tora so reserved or embarrassed before, but when you have a sister like Tami, it’s understandable.

I swear, these Chura sisters are going to drive me insane. The good kind of insane, though. I’m still debating if beating Tami’s cheeks that night was a good idea or not, but now that I’ve been let in, I don’t intend to close the door on myself for a very long time. Let’s just hope things between Tora and I won’t start to get messy. The bad kind of messy, by the way.
Roni

## Locations

* Courtyard - Thursdays
* Music Store - Weekend Mornings

## Entries

Roni has surprised me. She really isn’t the person I thought she’d be when I first met her. In fact, she’s trying to not be that person as much as possible. I like that her and I can talk about a variety of things and that I can forget I’m talking to someone several years younger than me. Now I just need to make sure she doesn’t think she needs to isolate herself from everyone else.
Light and Dream Events

## Light and Dream Events

This page contains all Light and Dream Events in Sensei Overnight

## Dream Events

### Chapter 1

* M.o.M (Dayai)
* The Kingdom's Chosen One (Tora)
* Keys For My Comfort (Kio)
* The Perfect Day (Eruka)

### Chapter 2

* Winter Skies and Butterflies (Tuli)
* Dreamscape(Mori)
* The Brother and The Sister (Sora)

## Light Events

### Chapter 1

* No Strings Attached
* You Be Her, and I'll Be You
* Endless Torture

### Chapter 2

* Stuck on Replay
* Sensei, Overnight!
* The Universe Sings To Me
Lina

## Locations

Lina currently doesn't have any locations

## Entries

This might sound weird, but Lina reminds me a little bit of Mori. But, where Mori pretty much despises me, Lina just doesn’t seem too bothered about anything. I still remember what she told me on our date, and I have the sneaky suspicion that she might be keeping an eye on me. I’ll try my best to stay on her good side.
Terminology

数学术语列表。

## 命题相关

terminology
nc.
术语
[例句]

axiom
nc.
公理
* ~ of choice选择公理

[例句]

proposition
nc.
命题
[例句]

theorem
nc.
定理
* Lagrange's ~拉格朗日定理

[例句]

corollary
nc.
推论
[例句]

definition
nc.
定义
[例句]

paradox
nc.
悖论
* Russell's ~罗素悖论

[例句]

proof
n.
证明，动词为 prove
[例句]

lemma
nc.
引理，复数 lemmas/lemmata
[例句]

hypothesis
n.
假设，复数 hypotheses
[例句]

principle
nc.
准则
* Cauchy ~ of convergence柯西收敛准则

[例句]

remark, note
nc.
注记
[例句]

example
nc.
例子
[例句]

## 结构词

if and only if, iff

当且仅当
[例句]

hence
adv.
因此
[例句]

for all

对任意，

∀

{\displaystyle \forall}

[例句]

there exists

存在，

∃

{\displaystyle \exists}

[例句]

assume
vt.
假设
[例句]

resp.

或
[例句]

verify
vt.
验证
[例句]

such that, s.t.

使得
[例句]

i.e.

即，源自拉丁文 id est
[例句]

with respect to

关于
[例句]

for convenience

方便起见
[例句]

be denoted

被记作
[例句]

inherit
vt.
继承
[例句]

arguing by contradiction

反证法
[例句]

be congruent to

与……一致
[例句]

with notation as above

记号同上
[例句]

as needed

即为所求
[例句]

Q.E.D.

证毕，源自拉丁文 quod erat demonstrandum
[例句]

e.g.

例如，源自拉丁文exempli gratia
[例句]

例句
--

参见
--