The maximum element of a partially ordered set. Terminology and designations. Special types of partially ordered sets

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Partially ordered set- a mathematical concept that formalizes intuitive ideas of ordering, arranging elements in a certain sequence. Informally, a set is partially ordered if it is specified which elements follow for which (which elements more which ones). IN general case It may turn out that some pairs of elements are not related by the relation " follows».

As an abstract example, we can give a set of subsets of a set of three elements $$ x, y, z$$ (boolean given set), ordered by inclusion relation.

Definition and examples

In order, or partial order, on the set $M$ called a binary relation $\varphi$ on $M$ (defined by some set $R_(\varphi) \subset M \times M$ ), satisfying the following conditions:

Reflexivity: $\forall a\; (a \varphi a)$
Transitivity: $\forall a, b, c\; (a \varphi b) \wedge (b \varphi c) \Rightarrow a \varphi c$
Antisymmetry : $\forall a, b\; (a \varphi b) \wedge (b \varphi a) \Rightarrow a = b$

A bunch of $M$ , on which the partial order relation is specified, is called partially ordered. To be completely precise, a partially ordered set is a pair $\langle M, \varphi \rangle$ , Where $M$ - a lot, and $\varphi$ - partial order relation on $M$ .

Terminology and notations

A partial order relation is usually denoted by the symbol $\leqslant$ , by analogy with the “less than or equal” relation on the set of real numbers. At the same time, if $a\leqslant b$ , then they say that the element $a$ does not exceed $b$ , or what $a$ subordinate $b$ .

If $a\leqslant b$ And $a \neq b$ , then they write $a< b$ , and they say that $a$ less $b$ , or what $a$ strictly subordinate $b$ .

Sometimes, in order to distinguish an arbitrary order on a certain set from a known “less than or equal” relation on the set real numbers, instead of $\leqslant$ And $<$ use special characters $\preccurlyeq$ And $\prec$ respectively.

Strict and non-strict order

A relation that satisfies the conditions of reflexivity, transitivity and antisymmetry is also called not strict, or reflexive order. If the condition of reflexivity is replaced by the condition anti-reflexivity(then the property of antisymmetry will be replaced by asymmetry):

$\forall a\; \neg (a \varphi a)$

then we get the definition strict, or anti-reflexive order.

If $\leqslant$ - loose order on the set $M$ , then the relation $<$ , defined as:

$a< b \; \overset{\mathrm{def}}{\Longleftrightarrow} \; (a \leqslant b) \wedge (a \neq b)$

is a strict order on $M$ . Back if $<$ - strict order, then attitude $\leqslant$ , defined as

$a\leqslant b\; \overset(\mathrm(def))(\Longleftrightarrow) \; (a< b) \vee (a = b)$

is a non-strict order.

Therefore, it makes no difference whether to define a loose order or a strict order on the set. The result will be the same structure. The only difference is in terminology and designations.

Examples

Top and bottom set

Special types of partially ordered sets

Linearly ordered sets

Let $\langle M, \leqslant\rangle$ is a partially ordered set. If in $M$ any two elements are comparable, then the set $M$ called linearly ordered. A linearly ordered set is also called completely orderly, or simply, ordered set. Thus, in a linearly ordered set for any two elements $a$ And $b$ one and only one of the relations holds: either $a, or a=b, or b .$

Also, any linearly ordered subset of a partially ordered set is called chain, that is, a chain in a partially ordered set $\langle M, \leqslant \rangle$ - a subset of it in which any two elements are comparable.

Of the above examples of partially ordered sets, only the set of real numbers is linearly ordered. The set of real-valued functions on an interval (given that $a), boolean \mathcal(P)(M) (at |M|\geqslant 2), natural numbers with a divisibility relation are not linearly ordered.$

In a linearly ordered set, the concepts of smallest and minimal, as well as largest and maximum, coincide.

Well-ordered sets

A linearly ordered set is called quite orderly, if each of its non-empty subsets has a smallest element. Such an order on a set is called in perfect order, in a context where it cannot be confused with complete order in the sense of .

A classic example of a well-ordered set is the set of natural numbers $\mathbb(N)$ . The statement that any non-empty subset $\mathbb(N)$ contains the smallest element, equivalent to the principle of mathematical induction. An example of a linearly ordered, but not completely ordered set is the set of non-negative numbers, naturally ordered $\mathbb(R)_(+) = $ x: x \geqslant 0$$ . Indeed, its subset $$x: x > 1$$ does not have a smallest element.

Well-ordered sets play an extremely important role in general set theory.

Complete partially ordered set- a partially ordered set that has bottom is the only element that precedes every other element and each directed subset of which has an exact upper bound. Complete partially ordered sets are used in λ-calculus and computer science, in particular, Scott's topology is introduced on them, on the basis of which a consistent model of λ-calculus and denotational semantics. A special case of a complete partially ordered set is a complete lattice - if any subset, not necessarily directed, has a supremum, then it turns out to be a complete lattice.

Ordered set $M$ is completely partially ordered if and only if each function $f\colon M\rightarrow M$ , monotonic with respect to the order ( $a \leqslant b \Rightarrow f(a) \leqslant f(b)$ ) has at least one fixed point: $\exists _(x \in M) f(x)=x$ .

Any set $M$ can be turned into a complete partially ordered one by highlighting the bottom $\bot$ and determining the order $\leqslant$ How $\bot \leqslant m$ And $m\leqslant m$ for all elements $m$ sets $M$ .

Theorems on partially ordered sets

Fundamental theorems about partially ordered sets include Hausdorff maximum principle And Kuratowski-Zorn lemma. Each of these theorems is equivalent to the axiom of choice.

In category theory

Every partially ordered set (and every preorder) can be considered a category, in which every set of morphisms $\mathrm(Hom)(A,B)$ consists of at most one element. For example, this category could be defined like this: $\mathrm(Hom)(A,B)=$(A,B)$$ , If A ≤ B(and the empty set otherwise); rule for the composition of morphisms: ( y, z)∘(x, y) = (x, z). Each preorder category is equivalent to a partially ordered set.

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Notes

Literature

Alexandrov P. S. Introduction to set theory and general topology. - M.: Nauka, 1977. - 368 p.
Barendregt, Henk. Lambda calculus. Its syntax and semantics = The Lambda Calculus. Its syntax and semantics. - M.: Mir, 1985. - 606 p. - 4800 copies.
Kolmogorov A. N., Fomin S. V. Elements of the theory of functions and functional analysis. - 7th ed. - M.: Fizmatlit, 2004. - 572 p. - ISBN 5-9221-0266-4.
Hausdorff F. Set theory. - 4th ed. - M.: URSS, 2007. - 304 p. - ISBN 978-5-382-00127-2.
Gurov S.I. Boolean algebras, ordered sets, lattices: Definitions, properties, examples. - 2nd ed. - M.: Librocom, 2013. - 352 p. - ISBN 978-5-397-03899-7.

An excerpt characterizing a partially ordered set

The killed animal near Borodino lay somewhere where the hunter who ran away had left it; but whether he was alive, whether he was strong, or whether he was just hiding, the hunter did not know. Suddenly the groan of this beast was heard.
The groan of this wounded beast, the French army, which exposed its destruction, was the sending of Lauriston to Kutuzov’s camp with a request for peace.
Napoleon, with his confidence that it is not only good that is good, but what came into his head that is good, wrote to Kutuzov the words that first came to his mind and had no meaning. He wrote:

“Monsieur le prince Koutouzov,” he wrote, “j"envoie pres de vous un de mes aides de camps generaux pour vous entretenir de plusieurs objets interessants. Je desire que Votre Altesse ajoute foi a ce qu"il lui dira, surtout lorsqu" il exprimera les sentiments d"estime et de particuliere consideration que j"ai depuis longtemps pour sa personne... Cette lettre n"etant a autre fin, je prie Dieu, Monsieur le prince Koutouzov, qu"il vous ait en sa sainte et digne garde ,
Moscou, le 3 Octobre, 1812. Signe:
Napoleon."
[Prince Kutuzov, I am sending you one of my general adjutants to negotiate with you on many important subjects. I ask Your Lordship to believe everything that he tells you, especially when he begins to express to you the feelings of respect and special reverence that I have had for you for a long time. Therefore, I pray to God to keep you under his sacred roof.
Moscow, October 3, 1812.
Napoleon. ]

“Je serais maudit par la posterite si l"on me regardait comme le premier moteur d"un accommodation quelconque. Tel est l "esprit actuel de ma nation", [I would be damned if they looked at me as the first instigator of any deal; such is the will of our people.] - answered Kutuzov and continued to use all his strength for that to keep troops from advancing.
In the month of the robbery of the French army in Moscow and the quiet stop of the Russian army near Tarutin, a change occurred in the strength of both troops (spirit and number), as a result of which the advantage of strength was on the side of the Russians. Despite the fact that the position of the French army and its strength were unknown to the Russians, how soon the attitude changed, the need for an offensive was immediately expressed in countless signs. These signs were: the sending of Lauriston, and the abundance of provisions in Tarutino, and information coming from all sides about the inaction and disorder of the French, and the recruitment of our regiments with recruits, and good weather, and the long rest of Russian soldiers, and the rest that usually arises in the troops as a result of rest. impatience to carry out the task for which everyone was gathered, and curiosity about what was happening in the French army, so long lost from sight, and the courage with which Russian outposts were now snooping around the French stationed in Tarutino, and news of easy victories over the French by the peasants and the partisans, and the envy aroused by this, and the feeling of revenge that lay in the soul of every person as long as the French were in Moscow, and (most importantly) the unclear, but arose in the soul of every soldier, consciousness that the relationship of force had now changed and the advantage is on our side. The essential balance of forces changed, and an offensive became necessary. And immediately, just as surely as the chimes begin to strike and play in a clock, when the hand has made a full circle, in the higher spheres, in accordance with a significant change in forces, the increased movement, hissing and play of the chimes was reflected.

The Russian army was controlled by Kutuzov with his headquarters and the sovereign from St. Petersburg. In St. Petersburg, even before receiving news of the abandonment of Moscow, a detailed plan for the entire war was drawn up and sent to Kutuzov for guidance. Despite the fact that this plan was drawn up on the assumption that Moscow was still in our hands, this plan was approved by headquarters and accepted for execution. Kutuzov only wrote that long-range sabotage is always difficult to carry out. And to resolve the difficulties encountered, new instructions and persons were sent who were supposed to monitor his actions and report on them.
In addition, now the entire headquarters in the Russian army has been transformed. The places of the murdered Bagration and the offended, retired Barclay were replaced. They thought very seriously about what would be better: to place A. in B.’s place, and B. in D.’s place, or, on the contrary, D. in A.’s place, etc., as if anything other than the pleasure of A. and B., it could depend on this.
At the army headquarters, on the occasion of Kutuzov’s hostility with his chief of staff, Bennigsen, and the presence of the sovereign’s trusted representatives and these movements, a more than usual complex game of parties was going on: A. undermined B., D. under S., etc. ., in all possible movements and combinations. With all these undermining, the subject of intrigue was mostly the military matter that all these people thought to lead; but this military matter went on independently of them, exactly as it should have gone, that is, never coinciding with what people came up with, but flowing from the essence of the attitude of the masses. All these inventions, crossing and intertwining, represented in the higher spheres only a true reflection of what was about to happen.
“Prince Mikhail Ilarionovich! – the sovereign wrote on October 2 in a letter received after the Battle of Tarutino. – Since September 2, Moscow has been in enemy hands. Your last reports are from the 20th; and during this entire time, not only has nothing been done to act against the enemy and liberate the capital, but even, according to your latest reports, you have retreated back. Serpukhov is already occupied by an enemy detachment, and Tula, with its famous and so necessary for the army factory, is in danger. From reports from General Wintzingerode, I see that the enemy 10,000th Corps is moving along the St. Petersburg road. Another, in several thousand, is also being submitted to Dmitrov. The third moved forward along the Vladimir road. The fourth, quite significant, stands between Ruza and Mozhaisk. Napoleon himself was in Moscow on the 25th. According to all this information, when the enemy fragmented his forces with strong detachments, when Napoleon himself was still in Moscow, with his guards, is it possible that the enemy forces in front of you were significant and did not allow you to act offensively? With probability, on the contrary, it must be assumed that he is pursuing you with detachments, or at least a corps, much weaker than the army entrusted to you. It seemed that, taking advantage of these circumstances, you could profitably attack an enemy weaker than you and destroy him or, at least, forcing him to retreat, retain in our hands a noble part of the provinces now occupied by the enemy, and thereby avert the danger from Tula and our other internal cities. It will remain your responsibility if the enemy is able to send a significant corps to St. Petersburg to threaten this capital, in which there could not be many troops left, for with the army entrusted to you, acting with determination and activity, you have all the means to avert this new misfortune. Remember that you still owe a response to the offended fatherland for the loss of Moscow. You have experienced my willingness to reward you. This readiness will not weaken in me, but I and Russia have the right to expect on your part all the zeal, firmness and success that your mind, your military talents and the courage of the troops led by you foretell to us.”
But while this letter, proving that a significant relationship of forces was already reflected in St. Petersburg, was on the way, Kutuzov could no longer keep the army he commanded from attacking, and the battle had already been given.
On October 2, Cossack Shapovalov, while traveling, killed one hare with a gun and shot another. Chasing a shot hare, Shapovalov wandered far into the forest and came across the left flank of Murat’s army, standing without any precautions. The Cossack, laughing, told his comrades how he almost got caught by the French. The cornet, having heard this story, reported it to the commander.
The Cossack was called and questioned; The Cossack commanders wanted to take this opportunity to recapture the horses, but one of the commanders, familiar with the highest ranks of the army, reported this fact to the staff general. Recently, the situation at army headquarters has been extremely tense. Ermolov, a few days before, having come to Bennigsen, begged him to use his influence on the commander-in-chief in order for an offensive to be made.
“If I didn’t know you, I would think that you don’t want what you’re asking for.” “As soon as I advise one thing, His Serene Highness will probably do the opposite,” Bennigsen answered.
The news of the Cossacks, confirmed by the sent patrols, proved the final maturity of the event. The stretched string jumped, and the clock hissed and the chimes began to play. Despite all his imaginary power, his intelligence, experience, knowledge of people, Kutuzov, taking into account the note from Bennigsen, who personally sent reports to the sovereign, the same desire expressed by all the generals, the desire of the sovereign assumed by him and the bringing together of the Cossacks, could no longer restrain inevitable movement and gave orders for what he considered useless and harmful - he blessed the accomplished fact.

The note submitted by Bennigsen about the need for an offensive, and the information from the Cossacks about the uncovered left flank of the French were only the last signs of the need to order an offensive, and the offensive was scheduled for October 5th.
On the morning of October 4, Kutuzov signed the disposition. Tol read it to Yermolov, inviting him to take care of further orders.
“Okay, okay, I don’t have time now,” said Ermolov and left the hut. The disposition compiled by Tol was very good. Just like in the Austerlitz disposition, it was written, although not in German:
“Die erste Colonne marschiert [The first column goes (German)] this way and that, die zweite Colonne marschiert [the second column goes (German)] this way and that way,” etc. And all these columns on paper they came to their place at the appointed time and destroyed the enemy. Everything was, as in all dispositions, perfectly thought out, and, as in all dispositions, not a single column arrived at its time and in its place.
When the disposition was ready in the required number of copies, an officer was called and sent to Ermolov to give him the papers for execution. A young cavalry officer, Kutuzov’s orderly, pleased with the importance of the assignment given to him, went to Ermolov’s apartment.
“We’ve left,” answered Yermolov’s orderly. The cavalry officer went to the general, who often visited Ermolov.
- No, and there is no general.
The cavalry officer, sitting on horseback, rode to another.
- No, they left.
“How could I not be responsible for the delay! What a shame! - thought the officer. He toured the entire camp. Some said that they saw Ermolov go somewhere with other generals, some said that he was probably home again. The officer, without having lunch, searched until six o'clock in the evening. Ermolov was nowhere and no one knew where he was. The officer quickly had a snack with a comrade and went back to the vanguard to see Miloradovich. Miloradovich was also not at home, but then he was told that Miloradovich was at General Kikin’s ball, and that Yermolov must be there too.
- Where is it?
“Over there, in Echkino,” said the Cossack officer, pointing to a distant landowner’s house.
- What’s it like there, behind the chain?
- They sent two of our regiments into a chain, there is such a revelry going on there now, it’s a disaster! Two musics, three choirs of songwriters.
The officer went behind the chain to Echkin. From afar, approaching the house, he heard the friendly, cheerful sounds of a soldier’s dancing song.
“In the meadows, ah... in the meadows!..” - he heard him whistling and clanking, occasionally drowned out by the shouting of voices. The officer felt joyful in his soul from these sounds, but at the same time he was afraid that he was to blame for not transmitting the important order entrusted to him for so long. It was already nine o'clock. He dismounted from his horse and entered the porch and entrance hall of a large, intact manor house, located between the Russians and the French. In the pantry and in the hallway footmen were bustling around with wines and dishes. There were songbooks under the windows. The officer was led through the door, and he suddenly saw all the most important generals of the army together, including the large, noticeable figure of Ermolov. All the generals were in unbuttoned frock coats, with red, animated faces and were laughing loudly, standing in a semicircle. In the middle of the hall, a handsome short general with a red face was smartly and deftly making a thrasher.
- Ha, ha, ha! Oh yes Nikolai Ivanovich! ha, ha, ha!..
The officer felt that by entering at this moment with an important order, he was doubly guilty, and he wanted to wait; but one of the generals saw him and, having learned why he was there, told Ermolov. Ermolov, with a frowning face, went out to the officer and, after listening, took the paper from him without telling him anything.
- Do you think he left by accident? - a staff comrade said to a cavalry officer about Ermolov that evening. - These are things, it’s all on purpose. Give Konovnitsyn a ride. Look, what a mess it will be tomorrow!

The next day, early in the morning, the decrepit Kutuzov got up, prayed to God, got dressed, and with the unpleasant consciousness that he had to lead a battle that he did not approve of, got into a carriage and drove out of Letashevka, five miles behind Tarutin, to the place where the advancing columns were to be assembled. Kutuzov rode, falling asleep and waking up and listening to see if there were any shots on the right, if things were starting? But everything was still quiet. The dawn of a damp and cloudy autumn day was just beginning. Approaching Tarutin, Kutuzov noticed cavalrymen leading their horses to water across the road along which the carriage was traveling. Kutuzov took a closer look at them, stopped the carriage and asked which regiment? The cavalrymen were from the column that should have been far ahead in ambush. “It might be a mistake,” thought the old commander-in-chief. But, having driven even further, Kutuzov saw infantry regiments, guns in their boxes, soldiers with porridge and firewood, in underpants. An officer was called. The officer reported that there was no order to move.

A set M in which an order relation is introduced, i.e. for some pairs of elements x, y an abstract relation x must follow the x introduced above (then it is called a strict order), the relation is connected as follows: or

Examples. 1. Set of real numbers with usual ordering; means the number is positive. In this case, for any pair of elements either y or

Set of all matrices with real elements; means that for everyone but . Obviously, there are “incomparable” matrices for which neither

The multiplicity of all continuous functions on a segment means that for all but

In this case there also exist pairs for which neither

The concept of partial ordering is important in combination with algebraic structures (eg the Abelian group), or algebraic and topological (in the theory of partially ordered linear spaces). Partial ordering in cybern. systems often have the character of hierarchical subordination. The simplest model of such a subordination is the relation of subordination between the faces of a simplex: it means that the face x is a proper face of the face y.

If M - Ch.y. m. with order then putting a -K b if and only if we define a new order on M. The resulting Ch. dual (or dual) to M. For any statement about Ch. m. there is a dual statement obtained by replacing the symbol. For example, the lower cone of the substitution of A in Ch. m. M is determined by the condition a for all a the upper cone by the condition: for all The element is maximum if or minimum if a. Element a in Ch. u. greatest (or one) if a for all . The smallest element (or zero) is defined dually. Of course, every largest (smallest) element is maximal (minimal), but not vice versa. If among the elements of the lower cone other than a there is a largest element b, then a is said to cover b (or that b immediately precedes a, or a immediately follows b). If Ch. m. M has “0” and “1”, then the row where covers the so-called. compositional row.

In the study Ch. m. and their applications, the principle of duality is extremely useful: if any theorem about Ch. formulated in general logical terms and terms of order, then its dual theorem is also valid.

If for any elements x and y from Ch.y. m. M one and only one of three statements holds: then the set M is called. linearly ordered (or perfectly ordered, as well as a chain). Every minimal (maximal) element of a linearly ordered set is the smallest (largest). Generally speaking, subsets of a linearly ordered set have no minima. elements; for example, in a set ordered by the usual “less than” relation, the part does not have a minimum. element. If each part of M has a minimum. element, M. called. a completely ordered set. For example, the set of natural numbers is completely ordered, but the set Z of all integers is not. According to Zermelo’s theorem (1904), any set can be completely ordered, i.e., an order relation that has the property described above can be introduced into it. Ch.u. isomorphic if there exists a bijective mapping such that it follows that If M is partially ordered, then for any subset is a segment M. For two completely ordered multiplicities in M and N, one can show that either M is isomorphic to the segment N or is isomorphic to the segment M: if true then and another, then M is isomorphic to N. Isomorphism is an equivalence relation between completely ordered sets; equivalence classes are called ordinal (ordinal) numbers. denotes the ordinal number corresponding to M. For ordinal numbers, the relation is introduced if M is isomorphic to the segment N, but not N. A finite ordinal number is an equivalence class containing a segment of the natural series, with

natural ordering. The smallest infinite ordinal number co is the class containing the entire natural series with natural ordering. Ordinal numbers are important as a means of proof using the method of transfinite induction, which is a natural generalization of the usual method of complete induction. Let it be necessary to prove a proposition P(a), the formulation of which contains an arbitrary ordinal number a. The principle of transfinite induction is that if P (1) is true and the validity of for implies the validity of P (a), then P (a) is true for all a. This principle can be proven as a theorem within the framework of axiomatic theory. Its application requires a preliminary complete ordering of the set of objects for which the proposition is being proven, which leads to their transfinite numbering; such ordering is possible due to Zermelo's axiom of choice. Using transfinite induction, a number of important theorems in mathematics are proved, for example, the Hahn-Banach theorem in functional analysis. The construction of various mathematics is also important. objects using transfinite induction. The use of transfinite induction is often replaced by an approach based on Zorn's theorem. Let M be Ch.y. m., X; if for everyone, then in the so-called. majorant X. If every linearly ordered subset has a majorant, M is called. inductive. Zorn's theorem that every inductive ordered set has at least one maximal element is widely used in algebra, functional analysis and other areas of mathematics. A visual representation of this theorem is given by ordering the subsets of a given set “by embedding” means.

The proof using Zorn's theorem is that we are looking for max. a subset of a given set M that has some property, and then it is proven that the assumption leads to a contradiction; from this they conclude that the entire set of M has the required property.

Lit.: Alexandroff P. Diskrete Raume. "Mathematical collection", 1937, vol. 2, century. 3; Kantorovich L. V., Vulikh B. Z., Pinsker A. G. Functional analysis in semi-ordered spaces. M.-L., 1950 [bibliogr. With. 543-546]; Kurosh A. G. Lectures on general algebra. M., 1962 [bibliogr. With. 383-387]; Skornyakov L. A. Elements of the theory of structures. M., 1970 [bibliogr. With. 145]; Riguet J. Binary relations, closures, Galois correspondences. In the book; Cybernetic collection, v. 7. M., 1963; Bourbaki N. Elements of mathematics, part 1. Basic structures of analysis, book. 2. Set theory. Per. from French M., 1965. A. V. Gladky.

A binary relation on a set A is called antisymmetric if:

(a,v A) a f c c f a

A binary relation on a set A is called reflexive if:

A binary relation on a set A is called transitive if:

(a,b,cA) aв вc > а с

The (entire) divisibility relation on the set of natural numbers N is antisymmetric. In fact, if ab, ba, then there are natural numbers q 1 ,qN, such that a=bq 1, b=aq from which a=aq 1 q, that is, q 1 q=1. But,

q 1 ,qN, therefore q 1 =q=1, which implies that a = b.

A reflexive antisymmetric transitive binary relation on the set A is called an order relation (partial order) on the set A.

Set A with a partial order relation given on it? is called a partially ordered set and denotes< А; ? >.

In what follows, for convenience, we will use the abbreviation CHUM, which denotes a partially ordered set.

< N, ? >? ordinary non-strict inequality of numbers (in the school sense). Is it necessary to prove the transitivity, reflexivity and antisymmetry of this relationship?

a) a ? a ,(2 ? 2) - reflexivity,

b) if a? in, in? s, then a ? c, (3 ? 4, 4 ? 5 > 3 ? 5) - transitivity,

c) if a ? in, in? a, then a=в, (3 ? 3, 3 ? 3 > 3=3) - antisymmetry.

It follows that< N, ? >- CHUM.

a) The divisibility relation on the set of natural numbers N is reflexive, since every number is a multiple of itself, that is, since for any aN there is always a = a 1 (1N), this, in the meaning of the relation, we have a a. Therefore, it is reflexive.

b) If the first number is divisible by the second (i.e., a multiple of the second), and the second is a multiple of the third, then the first is a multiple of the third, which means the relation is transitive, i.e. if a in, in c, a, in, cN. This means that there are q,qN such that

Let us denote: q = qqN. We have

where qN, i.e. and c - by definition. Therefore, the relation is transitive.

c) The antisymmetry of the relation follows from the fact that two natural numbers that are multiples of each other are equal to each other, i.e. if av, va, then there exist q 1 ,qN such that

that is, q 1 q=1. But, q 1 ,qN, therefore q 1 =q=1, which means that a = b. Therefore antisymmetric.

Therefore there is a partial order and, therefore,< N, >- CHUM (partially ordered set).

The elements a, in ChUMA A are called incomparable and are written

if a ? in and in? A.

Elements a,in ChUMA A are said to be comparable if a ? in or in? A.

Partial order? on A is called linear, and the plant itself is linearly ordered or a chain if any two elements from A are comparable, i.e. for any a, in A, or a ? in or in? a.

< N, ? >, - are a chain. However<В(М) ; >,where B(M) is the set of all subsets of the set M or B(M) is called a Boolean on the set M, is not a chain, because not for any two subsets the set M is one subset of the other.

Let< А, ? >- arbitrary plague.

An element mA is called minimal if for any x A from the fact that x ? m follows x = m.

The meaning of this concept is that A does not contain elements strictly smaller than this element m. They say that x is strictly less than m and write x< m, если x ? m, но притом x ? m. Аналогично определяется максимальный элемент этого ЧУМ. Ясно, что если m, m- разные минимальные (максимальные) элементы ЧУМ, то m || m.

In the theory of partially ordered sets, the condition a? c is sometimes read as follows: element a is contained in element b, or element b contains element a.

Each element of a finite Plague contains a minimum element and is contained in a maximum element of this Plague.

Proof:

Let a be an arbitrary element of a finite plague S. If a is a minimal element, then by virtue of reflexivity, the lemma is proven. If A is not minimal, then there is an element a such that

If a is minimal, then everything is proven. If the element ane is minimal, then for some a we obtain

If a is minimal, then from (1), (2), due to transitivity, we conclude that a contains the minimal element a. If a is not minimal, then

Thus, at some nth step of reasoning the process will terminate, which is equivalent to the fact that element a is minimal. Wherein

A< а<< а< а< а

Due to transitivity, it follows that the element a contains a minimal element a. Similarly, the element a is contained in the maximal element. The lemma is proven.

Consequence.

The final plague contains at least one minimal element.

Now we introduce the concept of a finite plague diagram S, which is important for further presentation.

First, we take all minimal elements m, m, m in S. According to the corollary, such elements exist. Then in the partially ordered set

S = S (m, m, m),

which, like S, is finite, take the minimal elements, and consider the set

The elements of the “first row” m, m, m are represented by dots. A little higher, we mark the elements of the “second row” with dots, and connect the dots with segments if and only if m<

Next, we find the minimal elements of the Plague, depict them with dots of the “third row” and connect them with dots of the “second row” in the manner indicated above. We continue the process until all elements of a given Plague S are exhausted. The process is finite due to the finiteness of the set S. The resulting set of points and segments is called the diagram of the Plague S. In this case, a< в тогда и только тогда, когда от “точки” а можно перейти к “точки” в по некоторой “восходящей” ломаной. В силу этого обстоятельства, любое конечное ЧУМ можно отождествить с его диаграммой.

Here given by the CHUM diagram

S = (m, m, ), in which m< , m< , m< m< , m< m< , m< .

An element m is called the smallest element of the plague if for any x A there is always m ? x.

It is clear that the smallest element is minimal, but the converse is not true: not every minimal element is the smallest. There is only one smallest element (if any). The largest element is determined similarly.

This is a plague, the elements of which are incomparable in pairs. Such partially ordered sets are called antichains.

This is the chain with the smallest and largest element. Where 0 is the smallest element and 1 is the largest element.

Let M be a subset of a partial ordered set A. An element a A is called the infimum of the set M if a? x for any x M.

The greatest of all infimums of a set M, if it exists, is called the infimum of the set M and is denoted by inf M.

Let< А, ? >- arbitrary plague. An element cA is called the infimum of elements a,in A if c = inf(a,b).

Note 1.

Not in every plague there is an exact infimum for any two elements.

Let's show this with an example.

For (a;c),(d;e) there is no lower bound,

inf(a;в)=d, inf(в;c)=e.

Let us give an example of a plague, which has an exact infimum for any elements.

inf(a;в)=d, inf(a;d)=d, inf(a;0)=0, inf(a;c)=0, inf(a;e)=0,

inf(in;c)=e, inf(in;e)=e, inf(in;d)=d,

inf(c;e)=c, inf(c;0)=0, inf(c;d)=0,

inf(d;e)=0, inf(d;0)=0,

Definition: A partially ordered set in which an infimum exists for any two elements is called a semilattice.

Example 10.

Let us give an example of a plague, which is not a semilattice.

Let< N, ? >- linearly ordered set of natural numbers and e, eN. On the set N=N( e ,e) let us define a binary relation? , assuming that x? y if x,y N, where x ? y, or if x N, y ( e ,e). We also calculate by definition: e ? e ,e? e.

The diagram of this plague is as follows:

Any natural number n? e and n? e, but not in N largest element, therefore, N is a CHUM, but not a semilattice.

So, by its very definition, a semilattice is a model (like a set with a relation?). As we will now see, another approach to the concept of a semilattice is possible, namely, a semilattice can be defined as a certain algebra.

To do this, we introduce some additional algebraic concepts. As is known, a semigroup is a non-empty set with an associative binary algebraic operation defined on it.

An arbitrary semigroup is usually denoted S (semigroup).

Definition. An element eS is called idempotent if

e= e, that is e · e = e.

Example 11.

Semigroup< N; · >? has only one idempotent 1.

Semigroup< Z; + >? has a single idempotent 0.

Semigroup< N; + >? does not have idempotent, because 0N.

For any non-empty set X, as usual, denotes the set of all subsets of the set X - the Boolean of the set X. Semigroup<В;>- is such that each of its elements is idempotent.

A semigroup is called an idempotent semigroup or copula if each of its elements is idempotent. Thus, examples of a connective are any Boolean relative to a union.

Example 12.

Let X be an arbitrary set.

B is the set of all subsets of the set X.

B- is called a Boolean on the set X.

If X = (1,2,3) then

B = (O,(1),(2),(3),(1,2),(2,3),(1,3),(1,2,3)).

Since the intersection of two subsets of the set X is again a subset of X, we have the groupoid< В;>, moreover, it is a semigroup and even a connective, since

A B and A = AA = A.

In exactly the same way, we have the connection<; В > .

A commutative connection is called a semilattice.

Example 13.

Let X = (1,2,3), let's build a diagram< В; >.

Let us give examples of plagues, but not semilattices.

Example 14.

Plague with two lower faces e and d, which are not comparable with each other: e||d. Therefore, inf(a;c) does not exist.

Example 15.

A plague with two lower faces c and d, which are incomparable with each other: c||d. Therefore, inf(a;в) does not exist.

Let us give examples of semilattices.

Example 16.

Diagram:

inf(a;в)=в, inf(a;с)=с, inf(a;d)=d,

inf(in;c)=d, inf(in;d)=d,

Example 17.

It is a semilattice, because for any two elements there is an infimum, i.e.

inf(a;в)=в, inf(a;с)=с, inf(в;c)=с.

Theorem 1.

Let ~~- semi-lattice. Then ~~commutative connective, where~~~~

aв = inf (a,в) (*).

Proof:

It is necessary to prove that in ~~the following identities hold:~~

(1) x y = y x

(2) (x y) z = x (y z)

1) According to equality(*)

x y = inf (x,y) = inf (y,x) = y x

2) Let us denote a = (x y) z, b = x (y z)

Let us prove that a = b.

To do this it is enough to prove that

V? a (5) (due to antisymmetry)

Let's denote

с = x y, d = y z

In meaning, the exact lower bound between c and z

A? s, huh? z , c ? x,

therefore, due to the transitivity of a ? x.

Same thing, huh? y, i.e. a is the common lower bound for y and z. And d is their exact lower bound.

Therefore, a? d, but in = inf (x,d).

From the inequality a? x,a? d it follows that a is some common infimum for x and d, and b is their exact infimum, therefore,

A? in (4) it is proved.

(5) is proved in a similar way.

From (4) and (5), in view of antisymmetry, we conclude that a = b.

With this we proved the associativity of the operation ().

3) We have x x = inf (x,x) = x.

Equality is achieved through reflexivity: x? X.

That. constructed algebra ~~will be a commutative idempotent semigroup, i.e. commutative link.~~

Theorem 2.

Let ~~is a commutative idempotent semigroup, then a binary relation? on S, defined by equality~~

? = ((a,в) S?S | a·в = а),

is a partial order. At the same time, PLAGUE ~~is a semilattice.~~

Proof:

1) reflexivity?.

By condition ~~satisfies three identities:~~

(1) x= x

(2) x y = y x

(3) (x y) z = x (y z)

Then x x = x = x - by virtue of (1). Therefore x? X.

2) antisymmetry? .

Let x? y and y? x, then by definition,

(4) x y = x

(5) y x = y

hence, thanks to commutativity, we have x = y.

3) transitivity?.

Let x? y and y? z then, by definition,

(6) x y = x

(7) y z = y

We have x z = (x y) z x (y z) x y x

So x·z = x, that is, x? z.

Thus, we have CHUM ~~. It remains to show that for any (a, b)S there exists inf(a, b).~~

Let's take it arbitrary a, b S and prove that the element c = a·b is inf(a,b), i.e. c = inf(a,b).

Indeed,

c a = (a b) a a (a b) (a a) b a b = c,

Similarly, с·в = (а·в)·в а·(в·в) а·в = с,

So, c is the common infimum of (a, b).

Let's prove its accuracy.

Let d be some common infimum for a and b:

(8)d? a

(9)d? V

(10) d·a = d

(11) d·в = d

d c = d (a b) (d a) c d c d,

d·c = d, therefore d ? c.

Conclusion: c = inf(a,b).

Theorems 1 and 2 proved allow us to look at semilattices from two points of view: as CUMs, and as in algebra (idempotent commutative semigroups).

A concept that formalizes intuitive ideas of ordering, arrangement in a certain sequence etc. Informally speaking, a set is partially ordered if it is specified which elements follow (more etc.) for which ones. In this case, in the general case, it may turn out that some pairs of elements are not related by the “follows” relation.

As an abstract example, we can give a collection of subsets of a set of three elements $$ x, y, z$$ , ordered by inclusion relation.

As an example “from life,” we can cite a set of people ordered in relation to “being an ancestor.”

Definition and examples

In order, or partial order, on the set $M$ called a binary relation $\varphi$ on $M$ (defined by some set $R_(\varphi) \subset M \times M$ ), satisfying the following conditions:

Reflexivity: $\forall a\; (a \varphi a)$

Transitivity: $\forall a, b, c\; (a \varphi b) \wedge (b \varphi c) \Rightarrow a \varphi c$

Antisymmetry: $\forall a, b\; (a \varphi b) \wedge (b \varphi a) \Rightarrow a = b$

A bunch of $M$ , on which the partial order relation is specified, is called partially ordered(English) partially ordered set, poset). To be completely precise, a partially ordered set is a pair $\langle M, \varphi \rangle$ , Where $M$ - a lot, and $\varphi$ - partial order relation on $M$ .

Terminology and notations

A partial order relation is usually denoted by the symbol $\leqslant$ , by analogy with the “less than or equal” relation on the set of real numbers. At the same time, if $a\leqslant b$ , then they say that the element $a$ does not exceed $b$ , or what $a$ subordinate $b$ .

If $a\leqslant b$ And $a \neq b$ , then they write $a< b$ , and they say that $a$ less $b$ , or what $a$ strictly subordinate $b$ .

Sometimes, in order to distinguish an arbitrary order on a certain set from the known “less than or equal to” relation on the set of real numbers, instead of $\leqslant$ And $<$ use special characters $\preccurlyeq$ And $\prec$ respectively.

Strict and non-strict order

A relation that satisfies the conditions of reflexivity, transitivity and antisymmetry is also called not strict, or reflexive order. If the condition of reflexivity is replaced by the condition anti-reflexivity:
$\forall a\; \neg (a \varphi a)$
then we get the definition strict, or anti-reflexive order.

If $\leqslant$ - loose order on the set $M$ , then the relation $<$ , defined as:
$a< b \; \overset{\mathrm{def}}{\Longleftrightarrow} \; (a \leqslant b) \wedge (a \neq b)$
is a strict order on $M$ . Back if $<$ - strict order, then attitude $\leqslant$ , defined as
$a\leqslant b\; \overset(\mathrm(def))(\Longleftrightarrow) \; (a< b) \vee (a = b)$
is a non-strict order.

Therefore, it makes no difference whether to define a loose order or a strict order on the set. The result will be the same structure. The only difference is in terminology and designations.

Examples

$\vartriangleright$ As mentioned above, the set of real numbers $\mathbb(R)$ partially ordered by less than or equal to $\leqslant$ .

$\vartriangleright$ Let $M$ - the set of all real-valued functions defined on the interval , that is, functions of the form
f \colon \to \mathbb(R)

Let us introduce the order relation $\leqslant$ on $M$ in the following way. We'll say that $f\leqslant g$ , if for everyone $x\in$ the inequality is satisfied $f(x) \leqslant g(x)$ . Obviously, the introduced relation is indeed a partial order relation.

$\vartriangleright$ Let $M$ - some set. A bunch of $\mathcal(P)(M)$ all subsets $M$ (called Boolean), partially ordered by inclusion $M\subseteq N$ .

$\vartriangleright$ The set of all natural numbers $\mathbb(N)$ partially ordered with respect to divisibility $m\mid n$ .

Related definitions

Incomparable elements

If $a$ And $b$ are real numbers, then one and only of the relations holds:
a< b, \qquad a=b, \qquad b

If $a$ And $b$ are elements of an arbitrary partially ordered set, then there is a fourth logical possibility: none of the above three relations are satisfied. In this case the elements $a$ And $b$ are called incomparable. For example, if $M$ - set of real-valued functions on an interval , then the elements $f(x) = x$ And $g(x) = 1-x$ will be incomparable. The possibility of the existence of incomparable elements explains the meaning of the term "partially ordered set".

Min/max and smallest/largest elements

Main articles: Maximum (mathematics) , Minimum (mathematics)

Because a partially ordered set may contain pairs of incomparable elements, two different definitions are introduced: minimum element And smallest element.

Element $a\in M$ called minimal(English) minimal element), if the element does not exist $b< a$ . In other words, $a$ - minimum element, if for any element $b \in M$ or $b>a$ , or $b=a$ , or $b$ And $a$ incomparable. Element $a$ called the smallest(English) least element, lower bound (opp. upper bound) ), if for any element $b \in M$ there is inequality $b\geqslant a$ . Obviously, every smallest element is also minimal, but the converse is not true in general: the minimal element $a$ may not be the smallest if elements exist $b$ , not comparable to $a$ .

Obviously, if there is a smallest element in a set, then it is unique. But there may be several minimum elements. As an example, consider the set $\mathbb(N)\setminus $ 1 $ = $ 2, 3, \ldots $$ natural numbers without unity, ordered by divisibility relation $\mid$ . Here the minimum elements will be prime numbers, but the smallest element does not exist.

The concepts are introduced similarly maximum(English) maximum element) And the greatest(English) greatest element) elements.

Top and bottom edges

Let $A$ - a subset of a partially ordered set $\langle M, \leqslant\rangle$ . Element $u \in M$ called top edge(English) upper bound) $A$ , if any element $a \in A$ does not exceed $u$ . The concept is introduced similarly bottom edge(English) lower bound) sets $A$ .

Any element larger than some upper bound $A$ , will also be the upper bound $A$ . And any element smaller than some infimum $A$ , will also be the lower bound $A$ . These considerations lead to the introduction of the concepts smallest upper bound(English) least upper bound) And largest lower bound(English) greatest lower bound).

Special types of partially ordered sets

Linearly ordered sets

Main article: Linearly ordered set

Let $\langle M, \leqslant\rangle$ is a partially ordered set. If in $M$ any two elements are comparable, then the set $M$ called linearly ordered(English) linearly ordered set). A linearly ordered set is also called completely orderly(English) totally ordered set), or simply, ordered set. Thus, in a linearly ordered set for any two elements $a$ And $b$ one and only one of the relations holds: either $a, or a=b, or b .$

Also, any linearly ordered subset of a partially ordered set is called chain(English) chain), that is, a chain in a partially ordered set $\langle M, \leqslant \rangle$ - a subset of it in which any two elements are comparable.

Of the above examples of partially ordered sets, only the set of real numbers is linearly ordered. The set of real-valued functions on an interval (given that $a), boolean \mathcal(P)(M) (at |M|\geqslant 2), natural numbers with a divisibility relation are not linearly ordered.$

In a linearly ordered set, the concepts of smallest and minimal, as well as largest and maximum, coincide.

Well-ordered sets

Main article: Well-ordered set

A linearly ordered set is called quite orderly(English) well-ordered), if each of its non-empty subsets has the smallest element. Accordingly, the order on a set is called in perfect order(English) well-order).

A classic example of a well-ordered set is the set of natural numbers $\mathbb(N)$ . The statement that any non-empty subset $\mathbb(N)$ contains the smallest element, equivalent to the principle of mathematical induction. An example of a linearly ordered but not completely ordered set is the set of non-negative numbers $\mathbb(R)_(+) = $ x: x \geqslant 0$$ . Indeed, its subset $$x: x > 1$$ does not have a smallest element.

Well-ordered sets play an extremely important role in general set theory.

Theorems on partially ordered sets

Fundamental theorems about partially ordered sets include Hausdorff maximum principle And Kuratowski-Zorn lemma. These statements are equivalent to each other and essentially rely on the so-called axiom of choice (in fact, they are equivalent to the axiom of choice).

Notes

Literature

Alexandrov P. S. Introduction to set theory and general topology. - M.: "SCIENCE", 1977. - 368 p.

Kolmogorov A. N., Fomin S. V. Elements of the theory of functions and functional analysis. - 7th ed. - M.: “FIZMATLIT”, 2004. - 572 p. - ISBN 5-9221-0266-4

Hausdorff F. Set theory. - 4th ed. - M.: URSS, 2007. - 304 p. - ISBN 978-5-382-00127-2

see also

Lattice

Ordinal number

Pre-order

cs:Uspořádaná množinaeo:Partordohu:Részbenrendezett halmazko:부분순서 nl:Partiële orde oc:Relacion d"òrdre ro:Relaţie de ordine sl:Relacija urejenostizh:偏序关系

Notification: The preliminary basis for this article was a similar article in http://ru.wikipedia.org, under the terms of CC-BY-SA, http://creativecommons.org/licenses/by-sa/3.0, which was subsequently changed, corrected and edited.

Definition 1: Let be an order relation on the set
. Element
called smallest (largest) element of the set , if the condition is met:
(for the smallest element);
(for the largest element).
Definition 2: Let - order relation on a set
. Element
called minimum (maximum) element of the set , if the condition is met:
(for the minimum element);
(for the maximum element).
If the order on a given set is linear, then the concepts of the smallest and minimal element (largest and maximum) coincide. Thus, these concepts are different only in the case when the order on the set is not linear.
If in abundance there is a smallest element, then it will also be minimal. The opposite is not always true. In addition, in a certain set for a given order relation there can be many minimal elements, but not a single smallest one.
Example:
1) On a set
consider the attitude
. All prime numbers in the order entered will be minimal, but none of them will be the smallest.
2) The set of all subsets of a certain set has a single minimal element - the empty set. In the set of all nonempty subsets of some arbitrary set, the minimal elements are all singleton subsets. Finally, if a set is infinite, then the set of all its infinite subsets has no minimal elements at all.
By definition, an element of a partially ordered set is the smallest if it is smaller than all the elements. It is clear that if the smallest element exists, then it is unique. The largest element is determined similarly.
Consider the class of partially ordered sets satisfying the following equivalent among themselves conditions:
1) Minimum condition . Any non-empty subset

has at least one minimal in the set
element.
2) Condition for breaking descending chains . Any strictly decreasing chain of elements
partially ordered set
breaks off at the end point. In other words, for any decreasing chain
there is such an index , at which this chain stabilizes, i.e.
.
3) Inductance condition. All elements of a partially ordered set
have some property , if all minimal elements of this set have this property (if they exist) and if, from the validity of the property for all elements strictly preceding some element , the validity of this property for the element itself can be deduced .
Theorem 1: The conditions of minimality, breaking of descending circuits and inductance are equivalent to each other.
Proof: 1) First, we show that the inductance condition follows from the minimality condition. Indeed, let a partially ordered set
satisfies the minimality condition and let in it for some property the premises of the inductance condition are satisfied. If at the same time there are many
there are elements that do not have the property , then let will be one of the minimal among such elements (the existence of such an element provided by the minimality condition). Element cannot be minimal in the entire set
, which follows from the premise of the inductance condition, and since all elements strictly preceding the element , property already possess, then by the inductance condition the element must have this property. We have arrived at a contradiction.
2) Let us show that the condition for breaking the decreasing circuits follows from the inductance condition. Let a partially ordered set
satisfies the inductance condition. Let's apply this condition to the following property: element has the property , if every strictly decreasing chain of elements starting with an element , breaks off at the final place. Obviously, all minimal elements of the set have this property
, if they exist. On the other hand, let all elements strictly preceding the element , have the property . This means that all descending chains starting from elements strictly preceding the element , break off at the final place, but then any strictly decreasing chain starting with the element , breaks off. From the inductance condition it follows that the property all elements of the set have
, i.e. in abundance
every strictly decreasing chain breaks.
3) The condition of minimality follows from the condition of breaking of decreasing chains. Suppose that the partially ordered set
does not satisfy the minimality condition, let there be a non-empty subset of it
, which has no minimum elements. Using the axiom of choice (see below), we mark one element in each non-empty subset of the set
, and then construct a sequence of elements
in the following way. As select the element marked in the subset
. If element , already selected and
, then as the next element
we take the element marked in non-empty (since
has no minimal elements) set of elements from
, strictly preceding . The constructed sequence is an infinite strictly decreasing chain. So there are many
does not satisfy the condition for breaking descending chains. The theorem is proven.
Definition 3: A partially ordered set is called structure or lattice , if any two-element subset in it has a tight upper bound
and the exact lower bound
. Exact upper bound
for elements And has the following properties:
, And
and
, If - any other upper limit of these elements. The exact lower bound is determined similarly.
Definition 4: The lattice is called distributive , if operations
And
are connected by distributive laws, i.e. the following relations are satisfied:
Comment: These ratios are called dual. Note that in a lattice one of these relations is a consequence of the other.
Definition 5: Zero And lattice unit its smallest and largest elements are called, if they exist.
Definition 6: In a lattice one can define the complement as follows: is a supplement for (A - addition for ), if at the same time
And
.
Definition 7: A distributive lattice with distinct 0s and 1s, in which every element has a complement, is called Boolean algebra.
Note that lattice theory and the theory of Boolean algebras are independent branches of algebra.
Definition 8: A linearly ordered set that satisfies the minimality condition (and therefore two other conditions equivalent to it) is called quite orderly .
An example of a well-ordered set is the set of natural numbers with natural order. Every subset of a completely ordered set is itself completely ordered. From the definition of a well-ordered set it follows that it has a single minimal element.
In a completely ordered set for every element there is an element immediately following . Element may not, however, have an immediately preceding element; in this case he will be called limit element.
When studying infinite sets one often has to use the following axiom, which is called axiom of choice.
Axiom: If given a set
, then there is a function , which associates each non-empty subset
specific element
this subset.
In other words, the function marks one element in each of the non-empty subsets of the set
.
The question of the logical foundations of this axiom and the legality of its use is one of the most difficult and controversial issues in the substantiation of set theory. Many of the theorems proved using the axiom of choice were completely contrary to clarity. Therefore, one of the prominent mathematicians of the 20th century, Bertrand Russell, spoke about this axiom as follows: “At first it seems obvious; but the more you think about it, the stranger the conclusions from this axiom seem; in the end you stop understanding what it means.”
There are a number of statements, each of which is equivalent to the axiom of choice. For example, the following theorems.
Cerielo's theorem: Any set can be completely ordered.
Hausdorff's theorem: Every chain of a partially ordered set is contained in some maximal chain.