Strict order relation. Strict order relation A strict linear order relation has the properties
The word "order" is often used in the most diverse issues. The officer gives the command: “Calculate in order of numbers”, arithmetic operations are performed in a certain order, athletes become in height, all leading chess players are arranged in a certain order according to the so-called Elo coefficients (an American professor who developed the system coefficients, which allows to take into account all the successes and failures of the players), after the championship, all football teams are arranged in a certain order, etc. planted a donkey not "!).
By arranging the elements of a certain set one after another, we thereby order them or establish some relationship between them. in a row. The simplest example is the natural order of natural numbers. Its naturalness lies in the fact that for any two natural numbers we know which of them follows the other or which of them is greater than the other, so we can arrange the natural numbers in a sequence so that the larger number will be located, for example, to the right of the smaller one: 1, 2, 3, ... . Of course, the sequence of elements can be written in any direction, and not just from left to right. The very concept of natural numbers already contains the idea of order. By establishing some relative arrangement of the elements of any set, we thereby set on it some binary order relation, which in each specific case can have its own name, for example, "be less", "be older", "contained in ", "follow", etc. Symbols for ordering can also be various, for example, Í, etc.
The main distinguishing feature of the order relation is that it has the property of transitivity. So, if we are dealing with a sequence of some objects x 1, x 2, ..., x n,... , ordered, for example, in relation to , then from what is performed x 1x 2... x n..., it should follow that for any pair x i , x j elements of this sequence is also performed x ixj:
For a pair of elements x ij in the relationship graph, we draw an arrow from the top x i to the top xj, i.e. from a smaller element to a larger one.
The order relation graph can be simplified by using the so-called Hasse diagrams. The Hasse diagram is constructed as follows. Smaller elements are placed below, and large ones are above. Since one such rule is not enough for the image, lines are drawn showing which of the two elements is larger and which is smaller than the other. In this case, it is enough to draw only lines for immediately following each other elements. Examples of Hasse diagrams are shown in the figure:
Arrows can be omitted in a Hasse diagram. The Hasse diagram can be rotated in the plane, but not arbitrarily. When turning, it is necessary to maintain the relative position (above - below) of the diagram vertices:
Attitude R in multitude X called relation of a strict order, if it is transitive and asymmetric.
A set in which a strict order relation is defined is called orderly. For example, the set of natural numbers is ordered by the relation "less than". But the same set is also ordered by another relation - “is divided by” and “greater”.
The graph of the "less than" relation in the set of natural numbers can be represented as a ray:
Attitude R V X is called the relation non-strict (partial) order, if it is transitive and antisymmetric. Every relation of nonstrict order is reflexive.
The epithet "partial" expresses the fact that perhaps not all elements of a set are comparable in this respect.
Typical examples of a partial order relation are "no more", "no less", "no older". The particle "not" in the names of relations serves to express their reflexivity. The relation "no more" coincides with the relation "less than or equal to", and the relation "not less" is the same as "greater than or equal to". In this regard, the partial order is also called lax in order. Often, a partial (non-strict) order relation is denoted by the symbol "".
The inclusion relation U between subsets of some set is also a partial order. Obviously, not any two subsets are comparable in this respect. The figure below shows a partial order by inclusion on the set of all subsets of the set (1,2,3). The arrows on the graph, which should point upwards, are not shown.
Sets on which a partial order is given are called partially ordered, or simply orderly sets.
Elements X And at partially ordered set are called compare, If Xat or atX. Otherwise, they are not comparable.
An ordered set in which any two elements are comparable is called linearly ordered, and the order is a linear order. Linear order is also called perfect order.
For example, the set of all real numbers with a natural order, as well as all its subsets, is linearly ordered.
Objects of the most diverse nature can be ordered hierarchically. Here are some examples.
Example 1: The parts of a book are ordered so that the book contains chapters, chapters contain sections, and sections consist of subsections.
Example 2. Folders in the computer file system are nested into each other, forming a branching structure.
Example 3. The relationship parents - children can be depicted in the form of the so-called family tree, which shows who is whose ancestor (or offspring).
Let on the set A given a partial order. Element X called maximum (minimum) element of the set A, if from the fact that Xat(atX), equality follows X= y. In other words, the element X is the maximum (minimum) if for any element at or it is not true that Xat(atX), or is performed X=y. Thus, the maximum (minimum) element is greater (less) than all other elements with which it is in relation.
Element X called largest (smallest), if for any atÎ A performed at< х (х< у).
A partially ordered set can have multiple minimum and/or maximum elements, but there cannot be more than one minimum and maximum element. The smallest (greatest) element is also the minimum (maximum), but the converse is not true. The figure on the left shows a partial order with two minimum and two maximum elements, and on the right - a partial order with the smallest and largest elements:
In a finite partially ordered set, there are always minimum and maximum elements.
An ordered set that has the largest and smallest elements is called limited . The figure shows an example of an infinite bounded set. Of course, it is impossible to depict an infinite set on a finite page, but it is possible to show the principle of its construction. Here loops near the vertices are not shown to simplify the drawing. For the same reason, the arcs that provide the display of the transitivity property are not shown. In other words, the figure shows a Hasse diagram of the order relation.
Infinite sets may not have a maximum, or a minimum, or both. For example, the set of natural numbers (1,2, 3, ...) has the smallest element 1 but no maximum. The set of all real numbers with natural order has neither the smallest nor the largest element. However, its subset consisting of all numbers X< 5 has a largest element (number 5) but no smallest element.
Let R be a binary relation on a set A.
DEFINITION. binary relation R on a set A is called an order relation on A or an order on A if it is transitive and antisymmetric.
DEFINITION. An order relation R on a set A is called non-strict if it is reflexive on A, i.e., for any of A.
An order relation R is said to be strict (on A) if it is antireflexive on A, i.e., for any of A. However, the antisymmetry of a transitive relation R follows from the fact that it is antireflexive. Therefore, we can give the following equivalent definition.
DEFINITION. A binary relation R on a set A is called a strict order on A if it is transitive and antireflexive on A.
Examples. 1. Let be the set of all subsets of the set M. The inclusion relation on the set is a non-strict order relation.
2. Relations on the set of real numbers are, respectively, a relation of strict and non-strict order.
3. The divisibility relation in the set of natural numbers is a relation of non-strict order.
DEFINITION. A binary relation R on a set A is called a preorder relation or a preorder on A if it is reflexive on and transitive.
Examples. 1. The ratio of divisibility in the set of integers is not an order. However, it is reflexive and transitive, which means it is a preorder.
2. The logical consequence relation is a preorder on the set of propositional logic formulas.
Linear order. An important special case of an order is a linear order.
DEFINITION. An order relation on a set is called a linear order relation or a linear order on if it is connected on , i.e. for any x, y from A
An order relation that is not linear is commonly referred to as a partial order relation or partial order.
Examples. 1. The relation "less than" on the set of real numbers is a relation of linear order.
2. The order relation accepted in the dictionaries of the Russian language is called lexicographic. The lexicographic order on the set of words in the Russian language is a linear order.
The word "order" is often used in a variety of issues. The officer gives the command: “Calculate in order of numbers”, arithmetic operations are performed in a certain order, athletes become in height, there is an order for performing operations in the manufacture of a part, word order in a sentence.
What is common in all cases when it comes to order? The fact that the word “order” has such a meaning: it means which element of this or that set follows which (or which element precedes which).
Attitude " X follows at» transitively: if « X follows at" And " at follows z", That " x follows z". In addition, this ratio must be antisymmetric: for two different X And at, If X follows at, That at does not follow X.
Definition. Attitude R on the set X called strict order relation, if it is transitive and antisymmetric.
Let us find out the features of the graph and the graph of strict order relations.
Consider an example. On the set X= (5, 7, 10, 15, 12) the relation R: « X < at". We define this relation by enumeration of pairs
R = {(5, 7), (5, 10), (5, 15), (5, 12), (7, 10), (7, 15), (7, 12), (10, 15), (10, 12), (12, 15)}.
Let's build its graph. We see that the graph of this relation has no loops. There are no double arrows on the graph. If from X the arrow goes to at, and from at- V z, then from X the arrow goes to z(Fig. 8).
The constructed graph allows you to arrange the elements of the set X in this order:
{5, 7, 10, 12, 15}.
In Fig. 6 (§ 6 of this chapter) columns VII, VIII are graphs of relations of a strict order.
Non-strict order relation
The relation "less than" in the set of real numbers is opposite to the relation "not less". It is no longer a strict order. The point is, at X = at, relations X ³ at And at ³ X, i.e. the relation "no less" is reflexive.
Definition. Attitude R on the set X called non-strict order relation, if it is reflexive, antisymmetric and transitive.
Such relations are unions of a strict order relation with an identity relation.
Consider the relation "no more" (£) for the set
X= (5, 7, 10, 15, 12). Let's build its graph (Fig. 9).
A non-strict order relation graph, unlike a strict order relation graph, has loops at each vertex.
On fig. 6 (§ 6 of this chapter) graphs V, VI are graphs of relations of non-strict order.
Ordered sets
A set may turn out to be ordered (they also say completely ordered) by some order relation, while another may be unordered or partially ordered by such a relation.
Definition. A bunch of X called orderly some order relation R if for any two elements x, y from X:
(X, at) Î R or ( y, x) Î R.
If R is a strict order relation, then the set X ordered by this relation under the condition: if X, at any two unequal elements of a set X, That ( X, at) Î R or ( y, x) Î R, or any two elements x, y sets X are equal.
It is known from the school mathematics course that number sets N , Z , Q , R ordered by the ratio "less than" (<).
The set of subsets of a certain set is not ordered by the introduction of an inclusion relation (U), or a strict inclusion relation (T) in the above sense, because there are subsets none of which is included in the other. In this case, the given set is said to be partially ordered by the relation Í (or Ì).
Consider the set X= (1, 2, 3, 4, 5, 6) and it has two relations "less than" and "divisible by". It is easy to check that both of these relations are order relations. The less than relation graph can be represented as a ray.
The graph of the relation "is divided by" can be represented only on a plane.
In addition, there are vertices on the graph of the second relation that are not connected by an arrow. For example, there is no arrow connecting the numbers 4 and 5 (Fig. 10).
The first relation X < at' is called linear. In general, if the order relation R(strict and non-strict) on the set X has the property: for any X, atÎ X or xRy, or yRx, then it is called a linear order relation, and the set X is a linearly ordered set.
If the set X of course, and consists of n elements, then the linear ordering X reduces to enumeration of its elements by numbers 1,2,3, ..., n.
Linearly ordered sets have a number of properties:
1°. Let a, b, c– set elements X, ordered by relation R. If it is known that aRv And vRc, then we say that the element V lies between the elements A And With.
2°. A bunch of X, linearly ordered by the relation R, is called discrete if between any two of its elements lies only a finite set of elements of this set.
3°. A linearly ordered set is called dense if for any two distinct elements of this set there is an element of the set lying between them.
An important type of binary relations is order relations. Strict order relation - a binary relation that is antireflexive, antisymmetric, and transitive:
designation - (A preceded b). Examples are
relations "greater than", "less than", "older", etc. For numbers, the usual notation is the signs "<", ">".
Non-strict order relation - binary reflexive, antisymmetric and transitive relation. Along with natural examples of non-strict inequalities for numbers, an example is the relationship between points in a plane or space "to be closer to the origin". Non-strict inequality, for integers and real numbers, can also be considered as a disjunction of equality and strict order relations.
If a sports tournament does not provide for division of places (i.e. each participant receives a certain, only eat / awarded place), then this is an example of a strict order; otherwise, non-strict.
Order relations are established on a set when, for some or all pairs of its.elements., the relation
precedence . Setting-for a set some order relation is called his "ordering, and "self. set as a result of this becomes orderly. Order relations can be introduced in different ways. For a finite set, any permutation of its elements "specifies some strict order. An infinite set can be ordered in an infinite number of ways. Only those orderings that have meaningful meaning are of interest.
If for the order relation R on the set .M and some different elements, at least one of the relations holds
aRb or b Ra , then the elements A And b called comparable otherwise - incomparable.
Completely (or linearly) ordered set M -
set on which the order relation is given, and any two elements of the set M comparable; partially ordered set- the same, but pairs of incomparable elements are allowed.
A linearly ordered set is a set of points on a straight line with the relation "to the right", a set of integer, rational, real numbers with respect to "greater than", etc.
An example of a partially ordered set is three-dimensional vectors, if the order is given as if
That is, if the precedence is satisfied in all three coordinates, the vectors (2, 8, 5) and (6, 9, 10) are comparable, and the vectors (2, 8, 5) and (12, 7, 40) are not comparable. This way of ordering can be extended to vectors of any dimension: vector
precedes the vector if
And done
Other examples of ordering can be considered on the set of vectors.
1) partial order: 
, If
Those. by the length of the vectors; vectors of the same length are incomparable.
2) linear order: 
, If a
The last example introduces the concept of alphabetical order.
Alphabet is a tuple of pairwise distinct characters called letters of the alphabet. An example is the alphabet of any European language, as well as the alphabet of 10 Arabic numerals. In a computer, the keyboard and some aids determine the alphabet of valid characters.
Word in the alphabetA - tuple of alphabet characters A. The word is written with alphabetic characters in a row, from left to right, without spaces A natural number is a word in the digital alphabet A formula is not always a word due to the non-linear arrangement of characters the presence of superscript (exponents) and subscript (indices of variables, bases of logarithms) symbols, fractional bar, signs radicals, etc.; however, by some conventions, it can be written into a string, which is used, for example, in computer programming (for example, the exponentiation sign is written as 2 multiplication signs in a row: 5**3 means the third power of the number 5.
Lexico-graphic (alphabetic) ordering - for various words in the alphabet with ordered
characters set ordering: if
possible presentation 
, at which either
(subword can be empty), or - empty subword
In this definition - a prefix (initial subword) that is the same for both words - or the first in a row on the left are different
characters, or - the last character in the word - tail
subwords.
Thus, the alphabetical ordering of words is determined by the first character that distinguishes them from the left (for example, the word KONUS precedes the word COSINUS, since they first differ in the third letter, and H precedes C in the Russian alphabet). It is also considered that the space character precedes any character of the alphabet - for the case when one of the words is a prefix of the other (for example, KOH and CONE)
Exercise. Check that the alphabetical ordering of natural numbers that have the same number of digits in decimal notation is the same as their ordering by magnitude.
Let A - partially ordered set. The element is called maximum V A, if there is no element for which A< b. Element A called greatest V A, if for any other than A item completed b<а-
are defined symmetrically minimum and least elements. The concepts of the largest and maximum (respectively, the smallest and minimum) elements are different - see. example in Fig.14. The set in Fig. 14a has the largest element R, it is also the maximum, there are two minimum elements: s and t there is no smallest. In Fig. 14b, on the contrary, the set having two maximum elements / and j , there is no greatest, minimum, it is the smallest - one: T.
In general, if a set has a largest (respectively, smallest) element, then only one (there may be none).
There can be several maximum and minimum elements (may not be at all - in an infinite set; in the final case, there must be).
Let's look at two more examples. - relation on the set N:
"Y divides X", or "X is the divisor of the number Y"(For example,
) is reflexive and transitive. Consider it on a finite set of divisors of the number 30.
The relation is a relation of partial order (non-strict)
and is represented by the following matrix of order 8, containing 31 characters
The corresponding scheme with 8 vertices must contain 31 bundles. . However, it will be more convenient for viewing if we exclude 8
links-loops depicting the reflexivity of the relation (diagonal elements of the matrix) and transitive links, i.e. bundles
If there is an intermediate number Z such that
(for example, a bunch because ). Then in the schema
there will be 12 ligaments (Fig. 15); the missing links are implied "by transitivity". The number 1 is the smallest and the number 30
the largest elements in . If we exclude from the number 30 and
consider the same partial order on the set , then
there is no largest element, but there are 3 maximum elements: 6, 10, 15
Now let's build the same scheme for the Boolean relation
(set of all subsets) of a three-element set
Contains 8 elements:
Check that if you match the elements a, b, c, the numbers 2, 3, 5, respectively, and the operations of union of sets are the multiplication of the corresponding numbers (i.e., for example, a subset corresponds to
product 2 5 = 10), then the relation matrix will be exactly
same as for relation ; schemes of these two relations with the described
abbreviations of loops and transitive connectives coincide up to notation (see Fig. 16). The smallest element is
And the biggest -
binary relations R on the set A And S on the set IN called isomorphic if between A and B it is possible to establish a one-to-one correspondence Г, in which, if (i.e.
elements are related R), then (images
these elements are related S).
Thus, partially ordered sets and are isomorphic.
The considered example admits a generalization.
The Boolean relation is a partial order. If
Those. a bunch of E contains P elements , then each
subset corresponds P-dimensional vector with
components , where is the characteristic function
sets A/ . The set of all such vectors can be considered as a set of points P-dimensional arithmetic space with coordinates 0 or 1, or, in other words, as vertices P-dimensional
unit cube, denoted by , i.e. cube with edges of unit length. For n = 1, 2, 3 indicated points represent respectively the ends of the segment, the vertices of the square and the cube - hence the common name. For /7=4, a graphic representation of this relationship is in Fig.17. Near each vertex of the 4-dimensional cube, the corresponding
subset of a 4-element set and four-dimensional
a vector representing the characteristic function of this subset. The vertices are connected to each other, corresponding to subsets that differ in the presence of exactly one element.
In Fig. 17, a four-dimensional cube is depicted in such a way that on one
level there are pairwise incomparable elements containing the same number of units in the record (from 0 to 4), or, in other words, the same number of elements in the represented subsets.
In Fig.18a,b - other visual representations of a 4-dimensional cube;
in Fig.18a the axis of the first variable OH directed upwards (intentional deviation from the vertical so that the various edges of the cube do not merge):
while the 3-dimensional subcube corresponding to X= 0 is located below, and for X= 1 - higher. On fig. 186 same axle OH directed from inside the cube to the outside, the inner subcube corresponds to X= Oh, and external - X= 1.
IN 
The material file shows an image of a 5-dimensional unit cube (p. 134).
Lecture Plan #14 Classification of binary relations
1. Classification of antisymmetric relations
2. Classification of reflexive relations
2.1. Quasi-order relations
2.2. Relations of non-strict partial order
2.3. Non-strict ordering relations
2.4. Poor quality order
2.5. Non-strict weak order
2.6. Non-strict order
3. Duality of relations of strict and non-strict order
4. Overview of the properties of various types of relationships
Classification of antisymmetric relations
Structure of graphs of acyclic relations
The structure of graphs of relations of qualitative order
Structure of relation graphs of weak order
Strict order relationships
A strict order (strict preference, strong order, strict linear order) is an antireflexive, transitive, weakly connected binary relation (12).
Strict order is a special case of weak order (strict partial preference) with an additional weakly connected condition.
Example: The relation "strictly less than" on the set of integers.
Classification of reflexive relations
Quasi-order relations
These binary relations make it possible to compare elements of a certain set, but not by similarity, but by arranging the elements of groups in a certain order, i.e. by partial ordering.
A quasi-order (non-strict partial preference) is a reflexive and transitive binary relation (3).
Example: "to be a brother" (Ivan-Peter, Andrey-Anna)
Properties of quasi-orders
1. The intersection of quasi-orders remains a quasi-order.
2. The symmetric part of the quasi-order has the properties of reflexivity, symmetry, and transitivity, and therefore is an equivalence relation. R c = R / R inv
3. With the help of this intersection, it is possible to select groups of variants that are equivalent to each other, then a non-strict partial order relation generated by the original relation can be established between the distinguished groups.
4. The asymmetric part of the quasi-order is a transitive and anti-reflexive relation = qualitative order.
Relations of non-strict partial order
A nonstrict partial order relation (4) is a relation that has the properties of reflexivity, antisymmetry, and transitivity.
A non-strict partial order is an antisymmetric quasi-order
Example: "be part" relation defined for sets (and their subsets)
Properties of non-strict partial orders
1. The intersection of nonstrict partial orders remains a nonstrict partial order.
2. The symmetric part of a nonstrict partial order is a diagonal.
3. The asymmetric part of a nonstrict partial order is a (strict) qualitative order.
4. In the theory of intelligent systems, an important role is played by partially ordered sets - domains together with non-strict partial order relations defined on them.
5. Partially ordered sets with the additional property that every pair of elements has upper and lower bounds are called lattices. Boolean algebras are a particular case of lattices.
Non-strict ordering relationships
A nonstrict ordering is a reflexive relation that has the weakly connected property (5).
A loose ordering can also be defined as a fully connected relation.
The non-strict ordering relation can be thought of as the result of combining some tolerance and dominance relations.
Properties of relations of non-strict partial ordering
1. The intersection and union of fully connected relations remains a fully connected relation.
2. The symmetric part of the non-strict partial ordering is the tolerance.
3. The asymmetric part of a nonstrict partial ordering is a dominance.
4. For fully connected relations, a necessary condition for transitivity is that the relation is negatively transitive.
5. For fully connected relations, the property of transitivity is a sufficient condition for the relation to be negatively transitive.
Relations of nonstrict qualitative order
A binary relation R is called a nonstrict qualitative order if it is negative and fully connected (6).
A nonstrict qualitative order is a negative nonstrict ordering.
The relation of nonstrict qualitative order can be represented as the result of combining some relations of tolerance and qualitative order.
Properties of relations of nonstrict qualitative order
1. The symmetrical part of the non-strict qualitative order is tolerance. NT?
2. The asymmetric part of a non-strict qualitative order is transitive, and therefore is a qualitative order relation.
3. Thus, the non-strict qualitative order relation can be represented as the result of the union of the tolerance and qualitative order relations generated by the original relation.
4. The dual relation has the properties of asymmetry and transitivity, therefore it is a relation of a qualitative order.
Relations of non-strict weak order
A nonstrict weak order is a fully connected transitive and negative transitive relation (7).
A nonstrict weak order is a fully connected transitive relation.
A non-strict weak order is a transitive non-strict order.
Properties of relations of non-strict weak order
1. The symmetric part of a nonstrict weak order is an equivalence.
2. The asymmetric part Rac of a nonstrict weak order is transitive, and therefore is a relation of qualitative order.
3. Thus, a non-strict weak order relation can be represented as the result of the union of the equivalence and weak order relations generated by the original relation.
4. A non-strict weak order can be represented as a set of partially ordered layers, each of which is an equivalence class.
Relations of non-strict (linear) order
A non-strict order (non-strict linear order) is an antisymmetric, transitive, fully connected binary relation (8).
A non-strict order is an antisymmetric non-strict weak order.
A non-strict order is an anti-symmetric non-strict order.
Properties of relations of non-strict linear order
1. The symmetric part of a non-strict order is a diagonal.
2. The asymmetric part R ac of non-strict order is transitive and weakly connected, and therefore is a relation of strict order.
3. The dual relation has the properties of asymmetry, negativity and weakly connectedness; therefore, it is a relation of strict order. In addition, it coincides with R ac.
4. Thus, the non-strict order relation can be represented as the result of the union of the diagonal and the strict order generated by the original relation.
Duality of relations of strict and nonstrict order
An overview of the properties of different types of relationships
