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Order relationship

Lecture



The order relation is usually denoted by £. The notation X £ Y means that the pair ( X, y ) belongs to the set A Ì M ´ M , which is an order relation in M , and X precedes Y (or Y follows X ).

The order relation has the properties:

· Reflexivity x £ X ;

· Transitivity; If X £ Y , and Y £ Z , then X £ Z ;

· Antisymmetry; If X £ Y , and Y £ X , then X = Y.

The set on which the order relation is defined is called Ordered , and it is said that the order is introduced by this relation.

If for any two elements X and Y of the set M, the relation X £ Y (or Y £ X ), then M - Completely (linearly) orderly .

For example , a set of natural or real numbers with a natural ratio of order £ are perfectly ordered.

In the general case, it may turn out that for some pairs ( X, y ) none of the ratios X £ Y and Y £ X does not hold. Such elements are called Incomparable , and the set M is called Partially ordered .

Examples of partial order are the relation “to be a divisor”, the inclusion relation Ì, etc. Thus, the inclusion relation on the set of subsets M I of a certain universe M is reflexive ( M I Ì M I ), transitive (if M I Ì M J , a M J Ì M K , then M I Ì M K ) and antisymmetrically (from M I Ì M J , and M J Ì M I follows M I = M J ), but among all possible subsets there are such that none of the ratios of M I Ì M N , and M N Ì M I It does not take place. Similarly, not all pairs of elements from the set of natural numbers are in the relation “to be a divisor”.

This note discusses one of the most commonly used binary relations, the relationship of order.

1. What is order?

Order plays a huge role in our lives. People throughout history have tried to streamline relations among themselves, the surrounding phenomena. Without order it is impossible to imagine human life. Try to imagine a society in which anarchy reigns or a dictionary in which words are arranged chaotically. But what is order? With some orderings, we are so accustomed that often they are simply not aware of, such as the grammatical order of words in a sentence. This article provides a formal definition of the order used in mathematics.

Definition 1.1 . A binary relation a on a set X is called an order relation if it is transitive :

  Order relationship

and antisymmetrically :

  Order relationship

Example 1.1 . Consider the attitude of "older" on many people. Obviously, it is transitive and antisymmetric, and, therefore, is an order relation.

Example 1.2 . The hierarchy of animals, built on the stages of evolution, is a relation of order (Fig. 1).

  Order relationship

Fig. 1. The main stages of the evolution of eukaryotic organisms

Definition 1.2 . The set X with the order relation defined on it   Order relationship is called an ordered set and is denoted by   Order relationship .

Ordered set   Order relationship with a small number of elements is visually represented as a directed graph. In this case, the elements of the set M are associated with the vertices of the graph (indicated by dots in the figure), and the elements of the relation a with arcs (lines with arrows).

For example, Figure 2 shows a directed graph representing the relation   Order relationship = {( a , a ), ( a , b ), ( a , c ), ( b , c)} on the set M = { a , b , c , d }.

  Order relationship

Fig. 2. Graph of an ordered set

You can set the order on the set in various ways. So, for example, in Figure 3 three ways of streamlining the four countries are shown.

Square Russia USA France England
Population USA Russia France England
Population density England France USA Russia

Fig. 3. Three ways to streamline

Figure 4 shows oriented graphs representing the relations "divided" and "less" on the set M = { 1 , 2 , 3 , 4 } of natural numbers.

  Order relationship

Fig. 4. Graphs of relations "divided" (a) and "less" (b) on the set { 1 , 2 , 3 , 4 }

Example. M = {0; 2; 5; 7}. M2 = {<0; 0>; <0; 2>; <0; 5>; <0; 7>; <2; 0>; <2; 2>; <2; 5>; <2; 7> ; <5; 0>; <5; 2>; <5; 5>;

<5; 7>; <7; 0>; <7; 2>; <7; 5>; <7; 7>}.

From the set M2, we select a subset R of those pairs in which a> b. R = {<2; 0>; <5; 0>; <5; 2>; <7; 0>; <7; 2>; <7; 5>;}. The set R defines the “more” relation for the elements of the set M. The graph corresponding to this relation is shown in Fig. 5. Each arrow is directed from a larger number to a smaller one.

  Order relationship   Order relationship


Fig.

2. Varieties of order relationships

Definition 2.1 . A relation of order a is called a relation of non - strict order on a set X , if a is reflexive :

  Order relationship

The ratio of non-strict order is usually denoted by the symbol   Order relationship . If a   Order relationship , then they say that "the element x precedes the element y " or " y follows x ".

Example 2.1 . Attitude   Order relationship on the set of real numbers is a nonstrict order relation.

Example 2.2 . On the set of subsets of some universal set U, the relation   Order relationship is a nonstrict order relation.

Example 2.3 . The relationship of subordination in the institution is not strict order on a set of employees of the institution.

Example 2.4 . Attitude   Order relationship ( m divides n ) on an arbitrary subset of natural numbers is a non-strict order. Figure 5 shows the graph corresponding to the ordered set   Order relationship

  Order relationship

Fig. 5. Graph of a non-strictly ordered set

Example 2.5 . The identity relation is both an equivalence relation and a non-strict order relation.

Definition 2.2 . Two elements   Order relationship are called comparable elements of an ordered set X if either   Order relationship either   Order relationship .

The incomparable elements in the ordered set of example 4 are, for example, elements 7 and 2, 2 and 3, 3 and 7.

Definition 2.3 . An order relation a is called a strict order relation on a set X if a is anti-reflexive :

  Order relationship .

A strict order relation is denoted by <.

Example 2.6 . Let f and g be functions with the same domains. We define the relation> as follows: f > g , if for any x from the domain of the function f ( x )> g ( x ). Obviously, this relation is a strict order relation.

For the functions f and g shown in Fig. 6, the relation f > g holds. The pairs of functions f and h , as well as g and h, are incomparable.

  Order relationship

Fig. 6. Three functions

Example 2.7 . The alphabetic order is a strict order relation on a set of letters.

Example 2.8 . Let a strictly order relation be given on the set X   Order relationship . How can you set a strict order relation on a set   Order relationship , that is, how to compare pairs of elements from the set X ? One of the possible options is as follows. On the set X, we define a relation   Order relationship condition:

  Order relationship

Attitude   Order relationship is a strict order.

Example 2.9. Another way to set a strict order on a set   Order relationship is as follows. We assume that the relation ( a , b ) holds   Order relationship ( c , d ) if

  Order relationship

This order relation is called lexicographic . In general, it is defined as follows. For the words v and w of the same length, we assume that v < w , if there exists a number k such that   Order relationship ,   Order relationship , ...,   Order relationship ,   Order relationship where   Order relationship - i - th letters of the words v and w, respectively. For words   Order relationship and   Order relationship ( k > 0 ) of different length is considered v < w , if   Order relationship or   Order relationship and w < v , if   Order relationship . This sorting method is used in dictionaries. In this order, for example:

childhood

Institute

12 <123 <4.

As already noted, it is convenient to represent ordered sets as graphs. Moreover, if   Order relationship is a strictly order relation, then the relation graph a does not contain cycles. The converse is also true: for any graph G without cycles, there exists a relation a of strict order such that the graph associated with this relation coincides with the transitive closure of the graph G. (The transitive closure of a graph G is a graph obtained from a graph G by adding arcs connecting each vertex a with vertices reachable from a .) Indeed, let G be a graph without contours. We define the relation on the set M of vertices of this graph   Order relationship if there is a path in the direction of the arcs leading from x to y . It is easy to see that due to the absence of cycles, the relation   Order relationship is a strict order.

Definition 2.4 . X set with binary relation   Order relationship called connected if for any two different elements x and y from X either   Order relationship either   Order relationship .

Definition 2.5 . A connected order relation on a set X is called a linear order relation.

Example 2.10 . The lexicographical order of words in a dictionary is a linear order.

Example 2.11 . The inclusion relation on a set of figures is not a linear order (Fig. 7).

  Order relationship

Fig. 7. Two incomparable figures

Example 2.12 . The attitude of "older" on many people is a linear order.

Example 2.13 . Consider a lot of people. They can be ordered in different ways, for example, by height (Fig. 8).

  Order relationship

Fig. 8. The rank

The ordering of the elements of the set X by mapping its elements into some ordered set Y is a fairly typical example of order determination.

The corresponding general method of ordering a set is as follows. Let an injective function be defined on a set X

f : X ® R ,

accepting real numeric values. Set the relation X by the condition: x < y if f ( x ) < f ( y ). Thus, a certain relation is f ( x ) < f ( x ) cannot be. The transitivity and antisymmetry of the relationship Finally, for any pair of different elements x , y from X, either f ( x ) < f ( y ) or f ( y ) < f ( x ) is true, since f is an injection. So the order The function f one-to-one maps our set X onto some subset of the set R of real numbers, so the relation x < y for any elements of the set X is equivalent to the inequality f ( x ) < f ( y ).

Definition 2.6 . Let a strict order relation X be given. The element x X X such that for every y from X that does not coincide with x , the relation x < y ( x > y ) holds is called the smallest ( largest ).

Definition 2.7 . Let a relation of strictly order X. Then the element   Order relationship is called minimal ( maximal ) in the ordered set < X , <> if there is no element y for which y < x (respectively, y > x ).

If, as usual, in the case of x < y, to draw an arrow from x to y , then in the relation graph the minimum element is the one into which the arrows do not enter, and the maximum element from which the arrows do not extend.

In the case of a linear strictly order, the minimal element x has the additional property that for every   Order relationship done x < y . Thus, for the case of linear orders, the concept of the minimal element coincides with the concept of the smallest element. In the general case, it may turn out that the element x is minimal, but is not in the relation x < y with any other elements. For example, in Figure 5, element 2 is minimal, but not comparable with elements 3 and 7.

To understand the difference between minimal and minimal elements, consider an example.

Example 2.14. Figure 9 shows a diagram of an ordered set, in which there are neither the smallest nor the largest elements, but at the same time there is exactly 1 minimal and exactly 1 maximal elements.

  Order relationship

Fig.9. Ordered set

Consider the finite strictly ordered sets in more detail.

Lemma 2.1. If a linear strict order is given on a finite (non-empty) set X , then there exists a smallest element, and it is unique.

Evidence. Take an arbitrary element   Order relationship . If a   Order relationship - the smallest element, then the existence of the desired element is proved. If not, then since   Order relationship , what   Order relationship . Again either   Order relationship - the smallest, or exists   Order relationship such that   Order relationship . We will continue this process. Suppose that n + 1 elements are already selected for which

  Order relationship .

By transitivity, it is clear that   Order relationship with i > j . Hence, due to anti-reflexivity, all the selected elements are not equal in pairs. Therefore, in view of the finiteness of the set X, the selection process must be interrupted in a finite number of steps. Element   Order relationship , chosen in the last step, will obviously be the desired one. So for any   Order relationship done   Order relationship . We show that this element is unique. Indeed, let there be another element   Order relationship such that, for everyone   Order relationship . Then simultaneously executed   Order relationship and   Order relationship that is impossible due to antisymmetry. The lemma is proved.

Theorem 2.1. Let a relation of linear strict order X. Then on X you can choose this numbering.   Order relationship what's the ratio   Order relationship will be performed if and only if i < j .

Evidence. Let be   Order relationship - the smallest element in the set X , chosen according to Lemma 1. Denote by   Order relationship lots of   Order relationship . Denote by   Order relationship smallest element of the set   Order relationship . It's clear that   Order relationship . Remove from   Order relationship element   Order relationship and the remaining set is denoted by   Order relationship . Its smallest element   Order relationship satisfies the condition:   Order relationship . The numbering procedure is already clear: going through all the elements of X sequentially using the indicated method, we will line them up:

  Order relationship ,

where p is the number of elements in X. By virtue of transitivity and antisymmetry, it is clear that   Order relationship if and only if i < j . The theorem is proved.

In essence, this theorem means that any linear strict order on a finite set X is equivalent to the usual order on some segment of the natural series.

The relation of order is directly related to the relation of dominance.

Definition 2.8. Let be   Order relationship is an ordered set, x and y are its elements. We say that x dominates y , if   Order relationship but there is no such element z О X that   Order relationship .

With the help of the dominance relation, you can introduce another visual way of representing ordered sets - Hasse diagrams . In the diagrams, each element is represented by a point in the plane, and if y dominates x , then x and y are joined by a segment, with the point corresponding to x located below y .

Consider three partial order relations and construct Hasse diagrams for them.

Example 2.15. Let A = { 1 , 2 , 3 }. On the set of all subsets of the set A, consider the relationship "to be a subset". A diagram of this ordered set is shown in Figure 10 (a).

Example 2.16. Let X = { 1 , 2 , 3 , 5 , 6 , 10 , 15 , 30 }. We introduce the relation "divided" on this set. The diagram of this ordered set is shown in Figure 10 (b).

Example 2.17. On the set X = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 }, consider the relation of linear order <. His diagram is shown in Figure 10 (c).

  Order relationship

Fig. 10. Hasse diagrams

Note that the Hasse diagrams of the first two relations are the same. This means that these partially ordered sets have the same structure, and this is different from the structure of the third ordered set, although it also contains eight elements. More precisely, such a common structure is defined by the concept of isomorphism.

Definition 2.9. Two ordered sets   Order relationship and   Order relationship are called isomorphic if there is a bijection   Order relationship preserving the relation of order. In other words,   Order relationship then and only if   Order relationship where   Order relationship and   Order relationship - отношения порядка, заданные на множествах X и Y соответственно.

Упорядоченные множества, рассмотренные в примерах 1 и 2 изоморфны.

Теорема 2.2. Всякое нестрого упорядоченное множество   Order relationship изоморфно некоторой системе подмножеств множества X , нестрого упорядоченной отношением включения.

Доказательство. Для каждого элемента   Order relationship рассмотрим множество   Order relationship . Then   Order relationship and   Order relationship - совокупность всех таких подмножеств. Докажем, что эта система подмножеств, нестрого упорядоченная отношением включения, изоморфна   Order relationship . Рассмотрим отображение   Order relationship такое, что   Order relationship . Then   Order relationship - биекция. Действительно, если   Order relationship , то, поскольку   Order relationship , в силу рефлексивности   Order relationship , имеем   Order relationship and   Order relationship . Аналогично получаем   Order relationship и в силу антисимметричности   Order relationship имеем a = b , т. е. отображение   Order relationship инъективно. Besides,   Order relationship сюръективно, так как у любого подмножества   Order relationship есть прообраз a . Докажем теперь, что отображение сохраняет отношение частичного порядка. Пусть   Order relationship . Тогда для любого   Order relationship из   Order relationship в силу транзитивности отношения   Order relationship следует   Order relationship , а значит, и   Order relationship . Если   Order relationship , то, поскольку   Order relationship , имеем   Order relationship , поетому   Order relationship .

Операции над отношениями порядка

Перейдем теперь к обсуждению вопросов, связанных с упорядочением по разным критериям. Пусть у нас имеется несколько отношений порядка. Как по ним построить новое отношение порядка?

Исходя из операций над множествами, мы можем определить ряд операций над отношениями. Далее будем полагать, что все отношения заданы на одном и том же множестве X .

Возьмем два отношения   Order relationship and   Order relationship . Каждому из них соответствует некоторое множество пар (подмножества   Order relationship and   Order relationship ).

Определение 3.1. Пересечением отношений   Order relationship называется отношение, определяемое пересечением соответствующих подмножеств. Ясно, что   Order relationship выполнено тогда и только тогда, когда одновременно выполнены соотношения   Order relationship и.

Пример 3.1. Пусть X - множество вещественных чисел, a - отношение "быть не меньше", b - отношение "быть строго больше". Then   Order relationship есть отношение "быть строго больше".

Определение 3.2. Объединением отношений   Order relationship называется отношение, определяемое объединением соответствующих множеств. Соотношение   Order relationship выполнено тогда и только тогда, когда выполнено хотя бы одно их соотношений   Order relationship or   Order relationship .

Пример 3.2. Если   Order relationship - отношение "больше" на множестве чисел, а   Order relationship - отношение "равно", то   Order relationship - это отношение   Order relationship .

Можно ввести операции непосредственно не сводящиеся к теоретико-множественным.

Определение 3.3. Обратным отношением называется отношение, определяемое условием:   Order relationship .

Пример 3.3. Пусть   Order relationship - отношение "делит", тогда   Order relationship - отношение "делится".

Определение 3.4. Произведением отношений называется отношение, определяемое следующим образом: соотношение   Order relationship равносильно тому, что существует такой элемент   Order relationship , для которого выполнены соотношения   Order relationship and   Order relationship .

Пример 3.4. Пусть   Order relationship - отношение "быть женой", а   Order relationship - отношение "быть отцом". Что означает в этом случае соотношение   Order relationship ? По определению существует такой z , что " x - жена z " и " z - отец y ". Другими словами, " x есть жена отца y ", т.е. " x - мать или мачеха y ".

Пример 3.5. Пусть   Order relationship - отношение "быть братом", а   Order relationship - отношение "быть родителем". Тогда произведение   Order relationship есть отношение "быть братом одного из родителей", т.е. "быть дядей".

Прежде чем мы перейдем к теоремам, относящимся к отношению порядка, рассмотрим несколько вспомогательных утверждений (ввиду их простоты мы приводим их без доказательства).

Лемма 3.1. Если отношения   Order relationship and   Order relationship рефлексивны, то рефлексивны и следующие отношения:

  Order relationship ,   Order relationship ,   Order relationship ,   Order relationship .

Лемма 3.2. Если отношения   Order relationship and   Order relationship симметричны, то симметричны и следующие отношения:

  Order relationship ,   Order relationship ,   Order relationship .

Лемма 3.3. Чтобы произведение   Order relationship симметричных отношений   Order relationship and   Order relationship было симметрично, необходимо и достаточно, чтобы отношения   Order relationship and   Order relationship коммутировали.

Лемма 3.4. Если отношения a и b - антисимметричны, то антисимметричны также и следующие отношения:   Order relationship ,   Order relationship .

Антисимметричность может не сохраняться при объединении и произведении.

Лемма 3.4. Если отношения   Order relationship and   Order relationship транзитивны, то транзитивны также следующие отношения:   Order relationship ,   Order relationship .

Из лемм 3.1, 3.2, 3.3, 3.4 вытекают следующие две теоремы.

Теорема 3.1. Если   Order relationship and   Order relationship - строгие порядки (нестрогие порядки), то пересечение   Order relationship также является строгим порядком (нестрогим порядком).

Замечание. Пересечение строгого и нестрогого порядка есть строгий порядок.

The property "to be a linear order" is not required to be maintained at the intersection. This is easiest to see from the following considerations. Let a be a linear order (strict or non-strict), then   Order relationship (or = E ). Therefore,   Order relationship on a set of more than one element it is not a linear order.

Theorem 3.2. If the relation  Order relationship is a strict (nonstrict, linear) order, then the relation  Order relationship is a strict (nonstrict, linear) order.

Combining orders in the general case is not an order. This is clearly seen in this example. Let be   Order relationship a linear weak order, then   Order relationship there is a relation of the same type. However, association   Order relationship is a complete relationship, and, therefore, is not an order. The following are without proof of the conditions under which the union of orders is an order.

Theorem 3.3. If a  Order relationship and   Order relationship - strict orders, the union   Order relationship is a strict order if and only if

  Order relationship .

Theorem 3.4. In order to combine  Order relationship weak orders  Order relationship and   Order relationship was lax order, it is necessary and sufficient to meet the conditions

  Order relationship ,

  Order relationship .

Произведение порядков   Order relationship также не обязано быть порядком. Это видно из того хотя бы, что для линейного нестрогого порядка   Order relationship произведение

  Order relationship

есть полное отношение. Достаточным условием является, например, такое.

Теорема 3.5. Если   Order relationship and   Order relationship - строгие порядки и выполнены соотношения

  Order relationship ,

  Order relationship ,

то   Order relationship - строгий порядок.

Conclusion

В данной лекции мы рассмотрели определения, разновидности и свойства отношений порядка. Однако, при этом не был затронут достаточно важный вопрос, как на практике строить отношения порядка? Этой проблемой занимаются такие разделы математики как теория выбора и принятия решений, теория голосования в больших и малых группах. Указанный вопрос мы рассмотрим в одной из следующих лекций.

LITERATURE

1. Пухначев Ю., Попов Ю. Математика без формул. - М.: АО "Столетие", 1995.

2. Birkhoff G., Barti T. Modern applied algebra. - M .: Mir, 1976.

3. Schrader Yu. A. Equality, similarity, order. - M .: Science, 1971.

4. Nefedov V.N., Osipova V.A. The Course of Discrete Mathematics. - M .: Publishing house of MAI, 1992.

Je. P. Emelchenkov, V. Je. Emelchenkov

THE BINARY RELATIONS. RELATIONS OF ORDER

Abstract. The relation of the order and its characteristics are considered in the a rticle.


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Discrete Math. Set theory. Graph theory. Combinatorics.

Terms: Discrete Math. Set theory. Graph theory. Combinatorics.