A completely ordered set of real numbers. Partially ordered set. Theorems on partially ordered sets

a set Pc defined on it by a binary relation that satisfies the following conditions: 4) in any non-empty subset ~ there exists an element a such that for all; thus V. u. m is a linearly ordered set that satisfies the minimality condition. The concept of V. at. m was introduced by G. Kantor. An example of V. u. m. serves as a naturally ordered set natural numbers. On the other hand, a segment of real numbers with natural order is not a V. m. Any subset of V. y. m. itself is quite orderly. Cartesian product of a finite number of V. u. m. is completely ordered by the lexicographic order relation. A linearly ordered set is completely ordered if and only if it does not contain a subset that is antiisomorphic (see Antiisomorphism of partially ordered sets) to the set of natural numbers. The smallest element of V. y. m. Rnaz. zero (and is denoted by 0). For any element there are many names. the initial segment of the set P. For any element a that is not the largest in P, there is an element immediately following it; it is usually denoted a+1. Element V. u. m., which does not have an immediate predecessor, is called limiting. Comparison theorem. For any two V. at. m. P1 and P2, one and only one of the following situations occurs: 1) P 1 is isomorphic to P 2, 2) P 1 is isomorphic to some initial segment of the set P 2, 3) P 2 is isomorphic to the initial segment of the set P1. Taking the axiom of choice sets as one of the axioms of the theory of sets, one can prove that on any non-empty set one can introduce an order relation that turns it into a variable equation. m. (i.e., any non-empty set can be completely ordered). This theorem, called Zermelo's theorem, is actually equivalent to the axiom of choice. Zermelo's theorem and the comparison theorem serve as a basis for comparing sets according to their cardinality. Ordinal types of V. at. m. called transfinites, or transfinite numbers. Lit.: Cantor G., "Math. Ann.", 1883, Bd 21, S. 51-8; Aleksandrov P.S., Introduction to the general theory of sets and functions, M.-L., 1948; Hausdorff F., Set Theory, trans. from German, M.-L., 1937; Bourbaki N., Set Theory, trans. from French, M., 1965; Kuratovsky K., Mostovsky A., Set theory, translated from English, M., 1970. B. A. Efimov, T. S. Fofanova.

View value Completely Ordered Set in other dictionaries

A bunch of- weight
great deal
abyss
abyss
dark
darkness-darkness
darkness of topics
a bunch
WHO
railway carriage
breakthrough
death
force
Synonym dictionary

Quite- adv. completely, completely, without lack, without measure. Measure it completely. | Excessively, in contentment, in abundance. They live well. | Everything without a trace, in full, completely, at all, in abundance.........
Dahl's Explanatory Dictionary

A bunch of- multiply, etc. see many.
Dahl's Explanatory Dictionary

A bunch of- multitudes, cf. (book). 1. units only Uncertain a large number of, number of something. workers. facts. I have heard many excellent singers in my life. Nekrasov. 2. Totality........
Ushakov's Explanatory Dictionary

Quite adv.- 1. Completely, completely, completely.
Explanatory Dictionary by Efremova

Not Quite Adv. Razg.— 1. Not fully.
Explanatory Dictionary by Efremova

Quite- adv. Completely, very, completely. be satisfied with the explanation. worthy man. Even if I don’t fully taste the joy. Pushkin.
Ushakov's Explanatory Dictionary

Quite- adv. Completely, completely, completely. V. is satisfied. V. ready. B. definite answer. V. enough.
Kuznetsov's Explanatory Dictionary

A bunch of- -A; Wed
1. A very large number, the number of someone, something. M. people. M. facts. Grow a lot of flowers. The evidence is abundant. Great m. examples (very........
Kuznetsov's Explanatory Dictionary

Reachable Set— Possible expected returns and standard deviation pairs of all portfolios that can be constructed from a given set of assets.
Economic dictionary

Feasible Set (or Opportunity Set)- a variety of portfolios that can be formed from securities considered by the investor.
Economic dictionary

A bunch of- a set of elements, parameters, combined according to some
attribute
Economic dictionary

Set of Admissible Solutions- the area within which it can be produced
choice of decisions limited by set goals and available resources.
Economic dictionary

Universal Set- , in mathematics - a SET containing all elements with a certain property. This is also called a hypothetical set, which should include all possible........
Scientific and technical encyclopedic dictionary

A bunch of- in mathematics, see Set theory.

Uncountable Set— concept of set theory; an infinite set whose cardinality is greater than that of a countable set. For example, the set of all real numbers is an uncountable set.
Large encyclopedic dictionary

Empty Set— concept of set theory; empty set - a set that does not contain a single element; is indicated? or 0. The concept of an empty set (similar to the concept of “zero”) arises........
Large encyclopedic dictionary

Countable Set— concept of set theory; A countable set is an infinite set whose elements can be numbered by natural numbers. The set of all rational numbers........
Large encyclopedic dictionary

Several or Many Necessary Reasons- a causal scheme that provides at least two reasons to explain what happens.
Sociological Dictionary

Several Or Many Satisfactory Reasons- a causal scheme that is triggered if, in the absence of any preliminary information, the situation provides the possibility of a variety of interpretations......
Sociological Dictionary

Class, Set (in Logic and Mathematics)- - a finite or infinite collection of objects, distinguished by a common characteristic (property or relationship), conceived as something whole. Objects that make up K.........
Philosophical Dictionary

Fuzzy Set— - a set with fuzzy boundaries, when the transition from elements belonging to the set to not belonging to the set occurs gradually, unsharply. In the classic........
Philosophical Dictionary

Normal Set— see: Contradiction in explicit definition.
Philosophical Dictionary

COMPLETELY- COMPLETELY, adv. Completely, completely. V. is satisfied.
Ozhegov's Explanatory Dictionary

A BUNCH OF- PLENTY, -a, cf. 1. A very large number, the number of someone or something. M. people. M. cases. There are plenty of stocks of all kinds. 2. In mathematics: a set of elements combined........
Ozhegov's Explanatory Dictionary

Let's consider the set, about some pairs of elements of which it is known that (i.e., the set is given order relation). The order relation can also be interpreted as a subset of the square of the set: in a table whose rows and columns correspond to the elements of the set, some cells are shaded - if the cell at the intersection of a column and a row is shaded, then .

An order relation is, of course, not any subset, it must satisfy the following properties:

1) for any ;

2) if and , then ;

3) if and , then .

Order relations are, for example, the usual comparison of numbers on a line (), nesting of sets (), the relation “divides” ( - divides).

Sometimes you want the order relation to fulfill some additional properties, for example, if there are no incomparable elements, i.e., about any two elements and you can say that either , or , then the ordering of the set is called linear ordering: all elements of a set can be arranged in ascending order.

Looking ahead a little, we will say that the ordering of the elements of a set is necessary, in particular, in order to be able to consider objects by induction: I want to be able to first consider the first element, prove a certain statement for it, and then, using the fact that this statement is true for the first elements, derive it for the th. For natural numbers, the proof of the principle of mathematical induction rests on the fact that any non-empty subset of natural numbers has smallest element.

From an arbitrary order relation and an arbitrary set one would like to fulfill a similar property: in any subset of the set under consideration there is a smallest element relative to the order relation under consideration. If a set is linearly ordered, and, in addition, in any of its subsets the smallest element can be identified, then it is called quite orderly.

Let's look at a few examples of well-ordered sets.

Empty set.

A bunch of .

A bunch of .

Note that these sets are ordered with respect to the membership relation (). It is not difficult to guess what a completely ordered set of three elements looks like for such an order relation:

The E set is obtained by combining the previous sets.

Definition. The sets constructed in this way are called natural numbers.

All these sets make up the set of natural numbers. Think about why the axiom of infinity is necessary for the existence of this set (see axiom of infinity).

Mikhail Raskin

There are several famous questions in set theory about whether some axiom implies another axiom (or hypothesis; an axiom is simply a hypothesis that is used by the vast majority). As in other areas of mathematics, unprovability can be demonstrated using a model in which the assumptions are true but the hypothesis is not true. To construct one of the most famous such examples, a model of set theory in which there is an intermediate power between the powers of the natural series and the real line, Cohen developed the method of forcing.

Victor Viktorov

Basic concepts, operations on sets, identities, properties of complement, De Morgan's rule, properties of symmetric difference; mapping (function), factor mapping, equivalence relation, barber's paradox; ordered sets, minimum, smallest, maximum and largest elements in an ordered set, majorant and minorant; axiom of choice, a completely ordered set.

Partially ordered set- a mathematical concept that formalizes intuitive ideas of ordering, the arrangement of 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, it may turn out that some pairs of elements are not related by the relation " follows».

As an abstract example, we can give a collection of subsets of a set of three elements (a Boolean of a given set), ordered by inclusion relation.

Definition and examples

In order, or partial order, on a set is a binary relation on (defined by some set) that satisfies the following conditions:

The set 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, where is a set, and is a partial order relation on .

Terminology and notations

A partial order relation is usually denoted by the symbol , similar to the less than or equal to relation on the set of real numbers. Moreover, if , then they say that the element does not exceed, or what subordinate .

If and, then they write and say that less, or what strictly subordinate .

Sometimes to distinguish random order on some set from the known relation “less than or equal to” on the set of real numbers, instead of and use Special symbols and correspondingly.

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):

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

If - loose order on the set, then the relation defined as:

is in strict order on . Conversely, if is a strict order, then the relation defined as

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

Let us introduce the order relation on as follows: , if the inequality . Obviously, the introduced relation is indeed a partial order relation.

Related definitions

Incomparable elements

If and are real numbers, then only one of the following relations holds:

If and 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 are called incomparable. For example, if is a set of real-valued functions on the interval , then the elements 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

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

The element is called minimal(English) minimal element) if the element does not exist. In other words, is a minimal element if for any element either , or , or and are incomparable. The element is called the smallest(English) least element, lower bound (opp. upper bound) ), if for any element the inequality . Obviously, every smallest element is also minimal, but the converse is not true in general: a minimal element may not be the smallest if there are elements that are not comparable to .

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 of natural numbers without one, ordered by the divisibility relation. 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 be a subset of a partially ordered set. The element is called top edge(English) upper bound) if any element does not exceed . The concept is introduced similarly bottom edge(English) lower bound) sets.

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

Top and bottom set

For an element of a partially ordered set upper set(English) upper set, upset) is the set of all elements preceded by ().

Complete partially ordered set(English) complete partial ordered, ω-complete partial ordered ) is 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 of calculations are built. 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.

An ordered set is completely partially ordered if and only if every function monotonic with respect to order () has at least one

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 $\backslash (\; x,\; y,\; z\backslash )$, 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 binary relation $\backslash varphi$ on $M$(defined by some set $R\_(\backslash varphi)\; \backslash subset\; M\; \backslash times\; M$), satisfying the following conditions:

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

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 $\backslash langle\; M,\; \backslash varphi\; \backslash rangle$, Where $M$- a lot, and $\backslash varphi$- partial order relation on $M$.

Terminology and notations

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

If $a\backslash leqslant\; b$ And $a\; \backslash neq\; b$, then they write , 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 $\backslash leqslant$ And use special characters $\backslash preccurlyeq$ And $\backslash prec$ 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:

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

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

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

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

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

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

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

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

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

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