Harbin Xingguang Middle School Mathematics

Ideal catalogue

Pinyin l ǐ xi m: ng

English 1.a dream; An ideal; Perfection 2. Ideal; Perfect; satisfactory

[Edit this paragraph] Interpretation

An abstract concept, set by people, refers to the life goals people hope to achieve and the prospects they pursue. Existence is the present, while ideal belongs to the future or the past, and it has a wide range, which can refer to concrete things or an abstract idea. The so-called ideal things are all based on people's understanding of the present.

Ideal and existence do not conflict. Ideal includes everything that exists now and in the past, and there is no way to deny its existence. But ideals are people-oriented, and it is meaningless to leave people's ideals for a while.

For example, the ideal of capitalism, the ideal lifestyle, the ideal life and the ideal social system.

Ideal, that is, reasonable imagination, is an ideology that only advanced animals such as human beings can fully embody.

The process and method of pursuing ideals: doing things in a down-to-earth manner, being honest and pursuing ideals.

[Edit this paragraph] About ideals.

Ideal is the goal of life and a possible imagination of the future. Ideal is the imagination of the future, but not all imagination is ideal.

Ideal is not only different from fantasy, but also different from fantasy and delusion. Ideal is correct imagination, which has outstanding characteristics different from fantasy, fantasy and delusion.

First, the ideal has objective inevitability. The objective inevitability of ideal lies in that ideal, as an imagination, correctly reflects the objective reality and the relationship between reality and the future, conforms to the law of things' change and development, and can be realized through hard work.

Second, the ideal is social. Ideal is a unique spiritual phenomenon of human beings, and ideal has distinct sociality. The sociality of ideal means that ideal is not the random imagination of isolated individuals who are divorced from society, but the imagination which is restricted and decided by society.

Third, the ideal has a class nature. In class society, ideals have distinct class nature. In a class society, because different classes have different social status and economic interests and pursue different goals, they form different ideals. People's class status and class interests determine that people's ideals must be branded with the class brand of class society. The ideal of unity among all classes does not exist.

[Edit this paragraph] Ideal category

Ideal is a phenomenon of social consciousness, and ideal is rich and colorful. It can be divided into different types from different angles.

First, ideals can be divided into individual ideals and group ideals according to the range of people they belong to. Personal ideal is an objective and inevitable imagination of the future. Group ideal is the same ideal of a certain group of people. Such as the ideals of political parties, groups and so on. Building socialism with China characteristics is the ideal of all the people in China, that is, the ideal of all the people in China.

Second, according to the length of the ideal struggle, ideals can be divided into long-term ideals and short-term ideals. A long-term ideal is an ideal that can only be realized through long-term struggle. For example, the establishment of a capitalist society is the long-term ideal of China producers. The near-term ideal is an ideal that can be realized in the near future.

Thirdly, according to the content of ideals, ideals can be divided into social ideals, life ideals, career ideals and quality ideals. Social ideal is people's vision for the future society. Social ideals include the foresight and assumption of the political system, economic system, scientific and cultural system and social outlook of the future society. Career ideal is people's pursuit and yearning for the future work department, the nature of work and the degree they have achieved in their career. The ideal of life is people's pursuit and yearning for future life. It includes the pursuit and yearning for material life such as food, clothing, housing and transportation, as well as spiritual life such as culture and entertainment, as well as marriage and family life. The ideal of quality is the goal of being a man and the pursuit and yearning for what kind of person to be. Among the above ideals, social ideal is the premise and foundation of other ideals. People always take the future society as the premise and foundation when designing their future career, life and life.

As an ideology, ideal is a concentrated expression of one's political stance, world outlook and outlook on life. People have advanced and reactionary political positions, scientific and unscientific world outlook, lofty and humble outlook on life. Therefore, people's ideals are divided into advanced and reactionary, lofty and humble. Advanced and lofty ideals are based on a correct understanding of objective laws, which conforms to the basic trend of social and historical development and the fundamental interests of the broad masses of the people. Otherwise, it is a reactionary and humble ideal.

[Edit this paragraph] Ideal new viewpoint

1, the ideal is human-centered, and it is meaningless to leave human ideals for a while. A person's ideals can have many kinds, which are varied and changeable in everyone's consciousness under the influence of cognition and values.

2. The correctness of the ideal. It is generally believed that a correct reflection of the objective reality and the relationship between reality and the future conforms to the law of the change and development of things and can be achieved through hard work. But based on the change of people's understanding of objective laws, objective laws can only be relative. For example, Newton's mechanics is different from Ai En Stein's theory of relativity, and ideals are also considered to be class-oriented. If there is no class, is there no ideal for people? Then how can we discuss the relationship between the present and the future?

3. Ideal is an ideology, and material determines consciousness. The realization of ideal is equal to the realization of material, which means that consciousness also exists objectively and is inevitable.

The ideal is to be as real as possible-to create a highly harmonious society.

The ideal is to realize the truth to the greatest extent and to make the society a highly harmonious society. This is the basic viewpoint of harmonious ideal view.

Ideal must be truth, otherwise it cannot be the direction of social development. Pursuing ideals is to make subjective knowledge, objective things, personal behavior, social practice and mutual relations conform to the objective reality that is conducive to promoting and realizing the harmonious development of human individuals, groups, the whole and all things in nature, or to say that they all conform to the truth.

Harmonious development and win-win situation are another expression of capitalist thought. * * * Capitalist society is a highly harmonious society. It is from the pursuit of * * * common prosperity to the pursuit of * * * common development, from the pursuit of the liberation of all mankind to the pursuit of harmonious development and prosperity between man and everything in the natural world. Standing on the shoulders of science, democracy and freedom, holding high the banner of "harmonious development and win-win for all" and creating a highly harmonious society-a big human family, a harmonious world and an ideal realm, is the highest ideal of mankind.

"Yes" in seeking truth from facts has three connotations: science, truth and ideal. What we explore, study and pursue is "yes". First, what is the true nature of objective things and what is the truth of the facts. This is the category of imperial examinations. The second is what the reasonable relationship of objective things should be, and what is the meaning of fairness, justice and rationality. This is the category of truth. Third, what may be the development future of objective things and what should be the ideal goal pursued by human beings. This is the most ideal category. In judicial practice, "taking facts as the basis, taking law as the criterion, and actively preventing crimes" embodies these three connotations.

Celebrity point of view

1, it is very important that high-sounding words and lofty ideals about the motherland should not become loud but empty words in our students' consciousness, and should not be eclipsed and unremarkable because of repeated repetition. Let children not talk about lofty ideals, let these ideals exist in the passion of young hearts, in the excited emotions and actions, and in the spirit of love and hate, loyalty and uncompromising.

-Suhomlinski

2. The secret of the success of moral education is that when a person is a teenager, he should show him the prospect of the whole world and personal life under the grand social life background.

-Suhomlinski

3. The core of moral education is to let everyone set a lofty purpose in life. ..... People seem to stride into the future every day, thinking about the future all the time and paying attention to the future. It is a long road from understanding social ideal to forming personal lofty purpose in life, which is education, first of all, emotional education.

-Suhomlinski

4. Thought is the foundation, and ideal is the green bud, on which the tree of human thought, activity, behavior, enthusiasm and passion grows.

-Suhomlinski

The instinctive difference between human spirit and animals is that we leave our beauty, ideals and beliefs about lofty and beautiful things to the next generation while reproducing offspring.

-Suhomlinski

6. Without faith, there is no worthy character and life; Without faith, there is no land worthy of the name. ...

-Whitman, American poet

7. You have to learn. If you don't learn, you can't be versatile, and if you don't want to learn, you can't succeed.

-Zhuge Liang

8. The feeling that beautiful things evoke in people's hearts is similar to our inner joy in front of our loved ones.

Chernyshevski

9. Everyone has certain ideals, which determine the direction and judgment of his efforts. In this sense, I have never regarded ease and happiness as the purpose of life itself-this ethical basis, which I call pigsty ideal. Illuminate my path, constantly give me new courage, let me face up to the ideal of life happily, that is, truth, goodness and beauty.

-Einstein

10, there is a life goal, a life goal, a period goal, a stage goal, a year goal, a month goal, a week goal, a day goal, an hour goal and a minute goal.

Leo Tolstoy

[Edit this paragraph] Famous sayings about ideals

1, the happiness and happiness of mankind lies in struggle, and the most precious thing is to fight for ideals.

The happiest thing in the world is to fight for the ideal.

3, the realization of the ideal depends only on doing, not on empty talk.

4. Men are born eight feet without ambition.

5, Tian Xingjian, the gentleman is constantly striving for self-improvement.

6. Where there is an ideal, there is strength. The dawn of victory is ahead. -Shen Baofeng

7, the heart is like a mirror, always wipe.

8. Ideal is the eye that reflects the beautiful soul.

9. Ideal is the thinking of seeking goals.

10. Ideal is the master of the world.

1 1. Only by knowing the road today can we plan the future clearly and wisely.

12. When the ideal is lost, the flower of youth will wither. Because the ideal is the light and heat of youth.

13, everyone has a certain ideal, which determines the direction of his efforts and judgment.

14, the ideal is not a delicate magnetic bowl, it can't be mended if it is broken; Ideal is a flower, which can bloom again when it fades.

15, the ideal is in ourselves, and at the same time, all kinds of obstacles that hinder us from realizing the ideal are also in ourselves.

16. aim like a mountain and walk like water. Not as good as mountain, not firm, not as good as water, not as good as winding path.

17. Ideal is the source of strength, the cradle of wisdom, the banner of charge, and the sword of cutting thorns.

18, the real joy of life is to die for a great goal.

19, a person's ideal ambition is often directly proportional to his ability.

20. The ideal of life is for the ideal life.

2 1, the higher a person's ideal, the purer his life.

22. If you are not indifferent, you will have no ambition, and if you are not quiet, you will be far away.

23. The highest ideal in life is to seek truth.

24. Living with an ideal, full of social interests and a clear purpose is the most beautiful and meaningful life in the world.

25. People need ideals, but they need human ideals that conform to nature, not supernatural ideals.

26. People who have no ideal in life are poor.

27. Your ideals and enthusiasm are the rudder and sail of your soul.

28. In the best world, everything is designed for a good purpose.

29. The ideal person should not only meet material needs, but also meet spiritual interests.

30. Life cannot be without ideals. There should be a healthy ideal, an ideal from the heart and an ideal from the people of the whole country.

3 1, the pleasure you enjoy in hope is far greater than the pleasure you actually enjoy in the future.

33. Ideal is beautiful, but without will, ideal is just a fleeting rainbow.

Great ideals can only be successfully realized through selfless struggle and sacrifice.

35. All those who die for a bright future are stone statues.

36. Apply your ideals to real things and you will have civilization.

37. Ideal has extraordinary charm for me.

38. People who kill their ideals are the worst murderers.

39. An ideal book is the key to wisdom.

40. Ideals, like the sun, attract all the mud on the ground.

4 1, the meditator has a goal, but the visionary does not.

42. People's wishes are endless, and people's strength is inexhaustible.

43. It is with hope that people can be relieved and live to the last moment of their lives.

44. The road to art is full of thorns, which is also a good thing. Ordinary people are discouraged, except the strong-willed.

45. Ideal is the guiding light. Without ideals, there is no firm direction. No direction, no life.

46. Be ambitious. -Zhuge Liang

47. Don't praise the good color, just leave a mouthful of dried Kun. -Wang Mian

48. People with incomplete feathers can't fly high. -Warring States policy

If the heart is unclear, there is no way to see, and if the will is irregular, there is no merit. -Lin Bu

If you want to rule the world, you can leave me alone in today's world. -Mencius

Andrew's straight ladder enveloped the blue sky, and Cao Changhui swept away the devil's breath. -Wang Ling

Fish that swallow a boat don't swim in tributaries; A swan flies high, but it doesn't collect sewage pools. -Liezi

If you are poor, you will be immune to it, and if you are good, you will help the world. -Mencius

Are you heartless? Thousands of miles of envy. -gu

Living in the world, carefree people don't think far; He who lives a leisurely life is not wide. -Confucius

* * * Don't be more ambitious at the beginning, and make a name for yourself later. -Du Xunhe

I am willing to fly for the world, I don't know how to get cold. —— Chen Gongyin

I want to shoot a night scene for Rand Park Jung Su. -Meng Jiao

Husband cleans Wan Li, who will clean the room? Liu Xiyi

The mouth is thin and the stone is thin, and the heart looks at the sea. -Han Yu

If I were Di Qing next year, there would be peach blossoms in the newspaper. -Huang Chao

I hope that the sword hanging around my waist can quickly pacify the border and serve my country. -Li Bai

Willing to ride the wind and waves. -Zong Xi

Try to climb to the top of the mountain: it dwarfs all the peaks under our feet. -Du Fu

The man fell to the ground and shot everywhere. Anneng Mountain is a beautiful garden. -Lu you

If you make up your mind, you will learn to think and follow it, then you will be prosperous or rich from tomorrow; If you are determined to win, you must fulfill your duties, be diligent and innovate. —— Wang Fuzhi

It's not easy to learn, and it's useless for a saint to return. -Chengdu (Cheng Hao, Franz Chen)

Earthworms take one point, and dragons swim for nine days. -Fang Xiaoru

Even with three different spirits, steelmaking has not changed much. -Wu Qi

Things are taboo and determined to be refined. -Liu Guo

Wan Li is thousands of miles away, enjoying the wonders brought by strange adventures. -Cha Shen Xing.

It takes three days to test the authenticity of jade, and it will take seven years to distinguish wood. -Bai Juyi

The sea can fill mountains and move mountains. Men are so ambitious. -Liu Guo

I am willing to calm the East China Sea, but I will not change my mind. There is no peace in the sea, and the heart is endless. -gu

I am eager to escape from all over the world, and I like Philip Burkart. -Tao Yuanming

I admire swans in the sky and compete for food. -Meng Haoran

If you can't plan vertically and horizontally, your generosity will remain. Wei Zhi

Show dry eventually becomes a building, fine steel don't make a hook. -Bao Zheng

Swear for white, sweep the world. -gu

Swallows play in the forest, swans know how to swim? -Cao Zhi

At the beginning of climbing, I felt that the sky was high and the clouds were wide, and I knew that the waves in the sea were slim. -wang pu

Traveling thousands of miles with a whip, how can you not love the warmth of a family? -Li Bai

Who pities turtles and cranes for thousands of years, but bears the heart of Peng Cheng and Wan Li? -Wen Tianxiang

Kunpeng has its own Tianchi storage. Who says it is too crazy to be bound? -Lu you

Eagle strikes the sky and the wind is strong, and Pengfei waves in spring. -Sima Guang

See a bed in the gap, see the center of the room in the window, see a room in the yard, and shine everywhere in the world. -Wei Yuan

Miracle Li Jing said that the intention is thousands of miles away. -Li Bai

I would like to whisper, hold high, and ask me to make an appointment. -Tao Yuanming

The drunken whale leans on the heavenly sword and laughs at the waves to help the boat in Sichuan. -Lu you

Sharpening a knife does not miss the woodcutter, and sharpening a knife does not miss the woodcutter. -Li Bai

I still like how spectacular it is in my eyes. When I look at the sea of clouds, I will see it blooming. -Lu you

When can the branches and leaves grow to the sky outside and go straight to thousands of feet to be upright? -Li Bai

I only have a dream of Yunshan in my life, and I can empty the mountains in one step. -Xu Xiake

Agitation and small success, and then enough to achieve great things; Don't tempt small profits, then you can make a big difference. -Historical records

If you don't fly, you will fly to heaven; If you don't sing, it will be a blockbuster. -Historical records

Huang Hu wiped the air at a stroke, who read the fan cage to repair the wing? -Yuan Haowen

The husband is weak-willed -Feng Menglong

There is no recovery in a thousand miles, and the old spirit can still cross Kyushu. -Lu you

Determined in this life. -Lu you

With lofty ideals and lofty ambitions, we young people are destined to achieve great success in the future. -Li Bai

Haste makes waste, and seeing small profits can't achieve great things. -The Analects of Confucius

There are no seeds at home, and men should be self-reliant. -Great.

Don't worry about your job, lest you win your ambition. -Zhu

The beginning of determination is to get rid of habits, smoke people, and never return until you are drunk. —— Wang Fuzhi

Digging a well as deep as a spring, sailing to help the people cross the river, hoping to assist the emperor. -Li Bai

Old people are born with needs and can't wait for the dust to grow. -Xue Xuan

Try to set it, and you will succeed in the end. —— Ouyang Xiu

Non-talent can't help his ambition, and non-ambition can't help his ambition. -Zhu

Only when you make a mistake will you regret it, and only when you are interested in fame and suffering will you donate. -Pu Songling

A courageous person must be ambitious. -Li Bai

The body can be in danger, but the will cannot be taken away. -The Analects of Confucius

If you don't have a strong will, you won't be wise. If you don't believe, you won't succeed. -Mozi

People without ambition are inhuman. -Ji Kang

People who made great achievements in ancient times not only had talent, but also had unremitting determination. -Su Shi

Water stirs the sound of stone, and people stir up ambition. -Qiu Jin

If you are ambitious, you will accomplish nothing if you don't stand up. -Mencius

How can a man lie in bed in the hands of his children when he dies in a field on the border and buries his ears in their boots when they die? Ma Yuan

Wan Li is still a close neighbor. -Cao Cao

Scholars can't help becoming Hongyi, they still have a long way to go. -The Analects of Confucius

Husband has ambition, which will be meritorious service. -Yang Jiong

People who face the wrong path are worried about walking fast, while those who look up to the mountains are determined to fly. -Fu Xuan

My husband's ambition is also handsome. -The Analects of Confucius

Stones can be broken, but they cannot be taken away; Dan can grind it, but he can't take its red color. -Lu Chunqiu

Although the head is long and low, it is indomitable. -Su Shi

Don't be an Qi Sheng in life, and get drunk and ride a long whale. -Lu you

High-spirited, thousands of miles away, generally do not go into the water. -Xin Qiji

Don't say Sang Yu is late, because it is still full of clouds. -Liu Yuxi

The heart is high with the bright moon, and the ambition is clean with the autumn frost. Li Shimin

Embrace wood and be born in it; The nine-story platform starts from the foundation soil; A journey of a thousand miles begins with a single step. -Lao Tzu

When you are old and strong, you would rather be moved; Poverty is strong, not falling into the sky. -Wang Bo

Like sailing against the current, my heart is like a rock. I hope my father will aim high and not be afraid of difficulties until he dies. -Li Shizhen

My Zhuang Xin didn't die at the same age. Even if I die, I can become a ghost wise man. -Lu you

For a thousand years, Jing Wei was calm, and on the third day, he ate cows in anger. -Lu you

Hold your breath, but don't be violent -"Mencius"

People's study is also without ambition, although there are many children. -Su Zhe

Those who aim high and can't reach it can have no regrets. How can they be laughed at? -Wang Anshi

Husband and wife have many tourists; The risks are far away, but the risks are few. However, the outlandish, magnificent and transcendent world outlook often lies in the perils and remoteness, while what people have is rare, where there is a will, there is a way.

To also. -Wang Anshi

If you don't have lofty aspirations, you must be firm. If you have a desire for peace, you will have leisure? Zhang Lei

Only if you don't stand up can you never stand up. -Zhu

Scholars must be determined that people who are so carefree today just look at what they have not done, but when they encounter anything, they will be beaten up. It's just a matter of perseverance. -Zhu

The ambition of a gentleman is related to more than one person and to the world for generations to come. The little man's concern is a once-in-a-lifetime concern. Don't be surprised. -Cheng Yi

If you don't have a will, there is nothing in the world. There is no skill that is not based on ambition. If you don't stand up, it's like a boat without a rudder and a horse without a title. What is it? —— Wang Shou

mercy

If you don't study hard, you won't have the determination. Wang Shouren

Determined to work hard is like planting trees, only the roots and buds are not dry, there are branches, leaves and flowers. Wang Shouren

If people are not far from their interests and absent-minded, they will learn nothing. -Zhang Zai

The three armed forces can win the handsome, but ordinary people can't win the ambition. -The Analects of Confucius

Dapeng rises with the wind one day and soars into Wan Li. -Li Bai

[Edit this paragraph] The poem "Ideal"

Author Liu Shahe

Ideal is a stone, knocking out a single spark;

Ideal is fire, lighting the extinguished lamp;

Ideal is a lamp that illuminates the road at night;

Ideal is the road that leads you to the dawn.

In the age of hunger and cold, the ideal is food and clothing;

In the era of food and clothing, ideal is civilization.

In chaotic times, the ideal is stability;

In a stable era, the ideal is prosperity.

Ideals are like pearls, one after another,

Throughout the ages, the future is endless.

Beautiful pearl chain, the backbone of history,

The past shines today, the present shines now, and the ancestors shine on their children.

Ideal is a compass to guide ships;

Ideal is a ship, carrying you to sea for a long voyage.

But the ideal is sometimes the arc of kissing between the sea and the sky.

It's beyond your tolerance and tortures your enterprising heart.

Ideal makes you smile and observe life;

Ideal makes you stubbornly resist fate.

Ideal makes you forget the early gray hair on your temples;

Ideal makes your head white and naive.

Ideal is an alarm clock to break your golden dream;

Ideal is soap to wash away your selfishness.

Ideal is not only an acquisition,

Ideal is a kind of sacrifice.

If ideals bring you honor,

That's just a by-product of it,

But more is misunderstood loneliness,

Laughter in loneliness, bitterness in laughter.

Ideal makes honest people often suffer misfortune;

Ideals keep the unfortunate people alive.

Ordinary people are great because they have ideals;

The ideal person is a "capitalized person".

There are always people in the world who have abandoned their ideals.

Ideals never abandon anyone.

Give new life to sinners, and ideal is the immortal grass that revives the soul;

Call the prodigal son back, the ideal is a loving mother.

The ideal has been defiled, so there is no need to resent it.

That's the devil testing your loyalty;

Ideal has been pickpocketed, don't cry,

Go get it back, and be careful in the future!

The hero lost his ideal and became a mediocre man.

Exaggerated the achievements of the year;

Mediocre people have lost their ideals and lived all their lives.

Ridiculously cursing the immediate environment.

Ideal flowering, peaches and plums should bear sweet fruit;

The ideal bud, elm poplar will have thick shade.

Please get on the ideal horse and start here with a whip.

Spring is just on the road and the sky is sunny.

The ideal of Liushahe is both sacrifice and gain. What does this mean?

The poet's "ideal" is a lofty social ideal, an ideal of social progress and the interests of most people. It is a kind of acquisition to establish ideals, strengthen beliefs and have happiness after fighting for ideals. And all this is sometimes at the expense of personal interests. He wanted to give up his golden dream and selfish interests, so it was a sacrifice.

What does the word "Zhao" mean in "Ideal" written by Liushahe?

The word "Zhao" in "The past shines today, the present shines now, and the ancestors shine on their descendants"

The word "Zhao" means "guidance", which not only inherits the "wide" in "Through the ages, the future is endless", but also takes care of the "Zhao" in "Ideal is a lamp, illuminating the night road", which not only vividly expresses the ideal of ancestors guiding future generations, but also makes poetry closely linked and integrated.

Background of "Ideal" of Liusha River

Liushahe had a rough life, and encountered great events in the Republic of China, but he still persisted in his ideals and fought for them. However, 1957 was wrongly classified as a rightist. The Third Plenary Session of 1978+0 1 rehabilitated him, returned to work in the Federation of Literary and Art Circles, and finally realized his ideal.

ideal

The ideal of a chicken is to find a handful of chaff.

The duck's ideal is to find a fish pond.

They saw eagles flying in the air,

Guess loudly, make noise,

The chicken said, "It must be a granary built on a cloud!" " "

The duck said, "Obviously someone is casting a net in the Milky Way!" "

The eagle smiled: "Anyone who only stares at his box will never know the real ideal ..."

[Edit this paragraph] Symphonic poem "Ideal"

Composed by Liszt, completed at 1857. Adapted from a poem of the same name by German writer Schiller. 1857 "Goethe-Schiller Monument" was completed in Weimar. The author created this song and is going to play it at the unveiling ceremony of the monument.

Citing information Zhimin, edited by Guohua. Concise music dictionary. Harbin: Heilongjiang People's Publishing House. 1985. Page 397.

[Edit this paragraph] Ideal mathematical concept

If the following conditions are true, then the nonempty subset I of a specific poset (p, ≤) is ideal:

For all x in I, y ≤ x implies y in I (I is a lower closed set).

For all x and y in I, there is an element z in I, so x ≤ z and y ≤ z. (I is a directed set)

Although this is the ideal and most general method to define arbitrary posets, it was originally defined only for lattices. In this case, the following equivalent definitions are given: the nonempty subset I of lattice (p, ≤) is ideal if and only if it is a finite union (supremum) lower closed set, that is, for all x and y in I, we find that xy is also in I.

The ideal dual concept, that is to say, the concept obtained by exchanging all ≤ in reverse, is a filter. The term ordered ideal or ordered filter is sometimes used for any lower set or upper set. In this paper, only "ideal/filter" and "lower closed set/upper closed set" are used to avoid confusion.

If an ideal or filter is not equal to the whole set p, it is called a true ideal or a true filter.

The smallest ideal containing a given element P is the principal ideal. In this case, P is called the principal element of the ideal. The principal ideal p of principal p is given as.

[Edit this paragraph] Chinese and English phrases

A rough country (contrary to utopia, a place full of ugliness and misfortune)

Cato is a bumpy state (the opposite of utopia is full of ugliness and misfortune) [see dysto-pia]

Idealized ...; Regard ... as ... as an ideal person (or thing); Describe (or reflect) idealizedly.

idealization

Ideal, perfection

ideal

Idealism; daydreamer

idealist

Wake up; Disillusionment of ideal

Disillusionment

Fantasy; Utopia. Antonym of DYSTOPIA

Yunjing

idealism

Characterization; ideal

flag

An ideal country, an imaginary place (or situation)

Installment payment system

Ideal rewards and punishments; Ideal karma

Ideal reward and punishment

An ideal husband, Mr. Right

Mr.Right

Two healthy incomes, no children, and early retirement (new Hakkas have higher ideals or goals than Dink and Yuppies. Thinkers don't have children, but they want a good income and early retirement.

ideologist

American social ideal, American lifestyle, American culture

the American dream

Ideal, wish

dream

Idealism; aerial

idealist

Ideal; unreasonable

Plato's

(often plural) an ideal thing

ideal

Form an ideal, idealize