In mathematics , the dimension of a partially ordered set (poset) is the smallest number of total orders the intersection of which gives rise to the partial order. This concept is also sometimes called the order dimension or the Dushnik–Miller dimension of the partial order. Dushnik & Miller (1941) first studied order dimension; for a more detailed treatment of this subject than provided here, see Trotter (1992) .
45-468: [REDACTED] Look up realizer in Wiktionary, the free dictionary. Realizer may refer to: For its use in mathematics see Order dimension CA-Realizer , the programming language similar to Visual Basic created by Computer Associates Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with
90-448: A i preceding b i , which would contradict the incomparability of a i and b i in P . And conversely, any family of linear orders that includes one order of this type for each i has P as its intersection. Thus, P has dimension exactly n . In fact, P is known as the standard example of a poset of dimension n , and is usually denoted by S n . The partial orders with order dimension two may be characterized as
135-426: A crown graph . The illustration shows an ordering of this type for n = 4. Then, for each i , any realizer must contain a linear order that begins with all the a j except a i (in some order), then includes b i , then a i , and ends with all the remaining b j . This is so because if there were a realizer that didn't include such an order, then the intersection of that realizer's orders would have
180-458: A function from A {\displaystyle A} to B {\displaystyle B} that is both injective and surjective . Such sets are said to be equipotent , equipollent , or equinumerous . For example, the set E = { 0 , 2 , 4 , 6 , ... } {\displaystyle E=\{0,2,4,6,{\text{...}}\}} of non-negative even numbers has
225-401: A one-to-one correspondence between the elements of two sets based on a unique relationship. In 1891, with the publication of Cantor's diagonal argument , he demonstrated that there are sets of numbers that cannot be placed in one-to-one correspondence with the set of natural numbers, i.e. uncountable sets that contain more elements than there are in the infinite set of natural numbers. While
270-545: A one-to-one correspondence with a strict subset (that is, having the same size in Cantor's sense); this notion of infinity is called Dedekind infinite . Cantor introduced the cardinal numbers, and showed—according to his bijection-based definition of size—that some infinite sets are greater than others. The smallest infinite cardinality is that of the natural numbers ( ℵ 0 {\displaystyle \aleph _{0}} ). One of Cantor's most important results
315-881: A poset P is the least integer t for which there exists a family of linear extensions of P so that, for every x and y in P , x precedes y in P if and only if it precedes y in all of the linear extensions, if any such t exists. That is, An alternative definition of order dimension is the minimal number of total orders such that P embeds into their product with componentwise ordering i.e. x ≤ y {\displaystyle x\leq y} if and only if x i ≤ y i {\displaystyle x_{i}\leq y_{i}} for all i ( Hiraguti 1955 , Milner & Pouzet 1990 ). A family R = ( < 1 , … , < t ) {\displaystyle {\mathcal {R}}=(<_{1},\dots ,<_{t})} of linear orders on X
360-475: A realizer of a finite partially ordered set P if and only if, for every critical pair ( x , y ) of P , y < i x for some order < i in R . Let n be a positive integer, and let P be the partial order on the elements a i and b i (for 1 ≤ i ≤ n ) in which a i ≤ b j whenever i ≠ j , but no other pairs are comparable. In particular, a i and b i are incomparable in P ; P can be viewed as an oriented form of
405-725: A relationship between sets which compares their relative size. For example, the sets A = { 1 , 2 , 3 } {\displaystyle A=\{1,2,3\}} and B = { 2 , 4 , 6 } {\displaystyle B=\{2,4,6\}} are the same size as they each contain 3 elements . Beginning in the late 19th century, this concept was generalized to infinite sets , which allows one to distinguish between different types of infinity, and to perform arithmetic on them. There are two notions often used when referring to cardinality: one which compares sets directly using bijections and injections , and another which uses cardinal numbers . The cardinality of
450-417: A set may also be called its size , when no confusion with other notions of size is possible. When two sets, A {\displaystyle A} and B {\displaystyle B} , have the same cardinality, it is usually written as | A | = | B | {\displaystyle |A|=|B|} ; however, if referring to
495-434: A set": Assuming the axiom of choice , the cardinalities of the infinite sets are denoted For each ordinal α {\displaystyle \alpha } , ℵ α + 1 {\displaystyle \aleph _{\alpha +1}} is the least cardinal number greater than ℵ α {\displaystyle \aleph _{\alpha }} . The cardinality of
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#1732802162163540-437: A similar argument, N {\displaystyle \mathbb {N} } has cardinality strictly less than the cardinality of the set R {\displaystyle \mathbb {R} } of all real numbers . For proofs, see Cantor's diagonal argument or Cantor's first uncountability proof . In the above section, "cardinality" of a set was defined functionally. In other words, it
585-501: Is a bijection. This is no longer true for infinite A {\displaystyle A} and B {\displaystyle B} . For example, the function g {\displaystyle g} from N {\displaystyle \mathbb {N} } to E {\displaystyle E} , defined by g ( n ) = 4 n {\displaystyle g(n)=4n}
630-499: Is an injective function from N {\displaystyle \mathbb {N} } to P ( N ) {\displaystyle {\mathcal {P}}(\mathbb {N} )} , and it can be shown that no function from N {\displaystyle \mathbb {N} } to P ( N ) {\displaystyle {\mathcal {P}}(\mathbb {N} )} can be bijective (see picture). By
675-470: Is apparent by considering, for instance, the tangent function , which provides a one-to-one correspondence between the interval (−½π, ½π) and R (see also Hilbert's paradox of the Grand Hotel ). The second result was first demonstrated by Cantor in 1878, but it became more apparent in 1890, when Giuseppe Peano introduced the space-filling curves , curved lines that twist and turn enough to fill
720-401: Is called a realizer of a poset P = ( X , < P ) if which is to say that for any x and y in X , x < P y precisely when x < 1 y , x < 2 y , ..., and x < t y . Thus, an equivalent definition of the dimension of a poset P is "the least cardinality of a realizer of P ." It can be shown that any nonempty family R of linear extensions is
765-429: Is equal to the number of points on a plane and, indeed, in any finite-dimensional space. These results are highly counterintuitive, because they imply that there exist proper subsets and proper supersets of an infinite set S that have the same size as S , although S contains elements that do not belong to its subsets, and the supersets of S contain elements that are not included in it. The first of these results
810-499: Is equivalent to the statement that | A | ≤ | B | {\displaystyle |A|\leq |B|} or | B | ≤ | A | {\displaystyle |B|\leq |A|} for every A {\displaystyle A} and B {\displaystyle B} . A {\displaystyle A} has cardinality strictly less than
855-475: Is injective, but not surjective since 2, for instance, is not mapped to, and h {\displaystyle h} from N {\displaystyle \mathbb {N} } to E {\displaystyle E} , defined by h ( n ) = n − ( n mod 2 ) {\displaystyle h(n)=n-(n{\text{ mod }}2)} (see: modulo operation )
900-434: Is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers, that is, However, this hypothesis can neither be proved nor disproved within the widely accepted ZFC axiomatic set theory , if ZFC is consistent. Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line, but that this
945-407: Is no set whose cardinality is strictly between that of the integers and that of the real numbers. The continuum hypothesis is independent of ZFC , a standard axiomatization of set theory; that is, it is impossible to prove the continuum hypothesis or its negation from ZFC—provided that ZFC is consistent. For more detail, see § Cardinality of the continuum below. If the axiom of choice holds,
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#1732802162163990-405: Is realized by two linear extensions, then partial order Q complementary to P may be realized by reversing one of the two linear extensions. Therefore, the comparability graphs of the partial orders of dimension two are exactly the permutation graphs , graphs that are both themselves comparability graphs and complementary to comparability graphs. The partial orders of order dimension two include
1035-546: Is surjective, but not injective, since 0 and 1 for instance both map to 0. Neither g {\displaystyle g} nor h {\displaystyle h} can challenge | E | = | N | {\displaystyle |E|=|\mathbb {N} |} , which was established by the existence of f {\displaystyle f} . A {\displaystyle A} has cardinality less than or equal to
1080-458: Is the notion of k -dimension (written dim k {\displaystyle {\textrm {dim}}_{k}} ) which is the minimal number of chains of length at most k in whose product the partial order can be embedded. In particular, the 2-dimension of an order can be seen as the size of the smallest set such that the order embeds in the inclusion order on this set. Cardinality In mathematics , cardinality describes
1125-816: The cardinal number of an individual set A {\displaystyle A} , it is simply denoted | A | {\displaystyle |A|} , with a vertical bar on each side; this is the same notation as absolute value , and the meaning depends on context. The cardinal number of a set A {\displaystyle A} may alternatively be denoted by n ( A ) {\displaystyle n(A)} , A {\displaystyle A} , card ( A ) {\displaystyle \operatorname {card} (A)} , or # A {\displaystyle \#A} . A crude sense of cardinality, an awareness that groups of things or events compare with other groups by containing more, fewer, or
1170-474: The law of trichotomy holds for cardinality. Thus we can make the following definitions: Our intuition gained from finite sets breaks down when dealing with infinite sets . In the late 19th century Georg Cantor , Gottlob Frege , Richard Dedekind and others rejected the view that the whole cannot be the same size as the part. One example of this is Hilbert's paradox of the Grand Hotel . Indeed, Dedekind defined an infinite set as one that can be placed into
1215-683: The natural numbers is denoted aleph-null ( ℵ 0 {\displaystyle \aleph _{0}} ), while the cardinality of the real numbers is denoted by " c {\displaystyle {\mathfrak {c}}} " (a lowercase fraktur script "c"), and is also referred to as the cardinality of the continuum . Cantor showed, using the diagonal argument , that c > ℵ 0 {\displaystyle {\mathfrak {c}}>\aleph _{0}} . We can show that c = 2 ℵ 0 {\displaystyle {\mathfrak {c}}=2^{\aleph _{0}}} , this also being
1260-431: The series-parallel partial orders ( Valdes, Tarjan & Lawler 1982 ). They are exactly the partial orders whose Hasse diagrams have dominance drawings , which can be obtained by using the positions in the two permutations of a realizer as Cartesian coordinates. It is possible to determine in polynomial time whether a given finite partially ordered set has order dimension at most two, for instance, by testing whether
1305-774: The cardinalities of unions and intersections are related by the following equation: Here V {\displaystyle V} denote a class of all sets, and Ord {\displaystyle {\mbox{Ord}}} denotes the class of all ordinal numbers. We use the intersection of a class which is defined by ( x ∈ ⋂ Q ) ⟺ ( ∀ q ∈ Q : x ∈ q ) {\displaystyle (x\in \bigcap Q)\iff (\forall q\in Q:x\in q)} , therefore ⋂ ∅ = V {\displaystyle \bigcap \emptyset =V} . In this case This definition allows also obtain
1350-656: The cardinality of B {\displaystyle B} , if there exists an injective function from A {\displaystyle A} into B {\displaystyle B} . If | A | ≤ | B | {\displaystyle |A|\leq |B|} and | B | ≤ | A | {\displaystyle |B|\leq |A|} , then | A | = | B | {\displaystyle |A|=|B|} (a fact known as Schröder–Bernstein theorem ). The axiom of choice
1395-659: The cardinality of B {\displaystyle B} , if there is an injective function, but no bijective function, from A {\displaystyle A} to B {\displaystyle B} . For example, the set N {\displaystyle \mathbb {N} } of all natural numbers has cardinality strictly less than its power set P ( N ) {\displaystyle {\mathcal {P}}(\mathbb {N} )} , because g ( n ) = { n } {\displaystyle g(n)=\{n\}}
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1440-474: The cardinality of a finite set is simply comparable to its number of elements, extending the notion to infinite sets usually starts with defining the notion of comparison of arbitrary sets (some of which are possibly infinite). Two sets have the same cardinality if there exists a bijection (a.k.a., one-to-one correspondence) from A {\displaystyle A} to B {\displaystyle B} , that is,
1485-474: The cardinality of the set of all subsets of the natural numbers. The continuum hypothesis says that ℵ 1 = 2 ℵ 0 {\displaystyle \aleph _{1}=2^{\aleph _{0}}} , i.e. 2 ℵ 0 {\displaystyle 2^{\aleph _{0}}} is the smallest cardinal number bigger than ℵ 0 {\displaystyle \aleph _{0}} , i.e. there
1530-461: The comparability graph of the partial order is a permutation graph. However, for any k ≥ 3, it is NP-complete to test whether the order dimension is at most k ( Yannakakis 1982 ). The incidence poset of any undirected graph G has the vertices and edges of G as its elements; in this poset, x ≤ y if either x = y or x is a vertex, y is an edge, and x is an endpoint of y . Certain kinds of graphs may be characterized by
1575-407: The division of things into parts repeated without limit. In Euclid's Elements , commensurability was described as the ability to compare the length of two line segments, a and b , as a ratio, as long as there were a third segment, no matter how small, that could be laid end-to-end a whole number of times into both a and b . But with the discovery of irrational numbers , it was seen that even
1620-402: The infinite set of all rational numbers was not enough to describe the length of every possible line segment. Still, there was no concept of infinite sets as something that had cardinality. To better understand infinite sets, a notion of cardinality was formulated c. 1880 by Georg Cantor , the originator of set theory . He examined the process of equating two sets with bijection ,
1665-447: The manipulation of numbers without reference to a specific group of things or events. From the 6th century BCE, the writings of Greek philosophers show hints of the cardinality of infinite sets. While they considered the notion of infinity as an endless series of actions, such as adding 1 to a number repeatedly, they did not consider the size of an infinite set of numbers to be a thing. The ancient Greek notion of infinity also considered
1710-781: The order dimensions of their incidence posets: a graph is a path graph if and only if the order dimension of its incidence poset is at most two, and according to Schnyder's theorem it is a planar graph if and only if the order dimension of its incidence poset is at most three ( Schnyder 1989 ). For a complete graph on n vertices, the order dimension of the incidence poset is Θ ( log log n ) {\displaystyle \Theta (\log \log n)} ( Hoşten & Morris 1999 ). It follows that all simple n -vertex graphs have incidence posets with order dimension O ( log log n ) {\displaystyle O(\log \log n)} . A generalization of dimension
1755-411: The partial orders whose comparability graph is the complement of the comparability graph of a different partial order ( Baker, Fishburn & Roberts 1971 ). That is, P is a partial order with order dimension two if and only if there exists a partial order Q on the same set of elements, such that every pair x , y of distinct elements is comparable in exactly one of these two partial orders. If P
1800-966: The same cardinality as the set N = { 0 , 1 , 2 , 3 , ... } {\displaystyle \mathbb {N} =\{0,1,2,3,{\text{...}}\}} of natural numbers , since the function f ( n ) = 2 n {\displaystyle f(n)=2n} is a bijection from N {\displaystyle \mathbb {N} } to E {\displaystyle E} (see picture). For finite sets A {\displaystyle A} and B {\displaystyle B} , if some bijection exists from A {\displaystyle A} to B {\displaystyle B} , then each injective or surjective function from A {\displaystyle A} to B {\displaystyle B}
1845-500: The same number of instances, is observed in a variety of present-day animal species, suggesting an origin millions of years ago. Human expression of cardinality is seen as early as 40 000 years ago, with equating the size of a group with a group of recorded notches, or a representative collection of other things, such as sticks and shells. The abstraction of cardinality as a number is evident by 3000 BCE, in Sumerian mathematics and
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1890-467: The title Realizer . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Realizer&oldid=1084033293 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Order dimension The dimension of
1935-1081: The whole of any square, or cube, or hypercube , or finite-dimensional space. These curves are not a direct proof that a line has the same number of points as a finite-dimensional space, but they can be used to obtain such a proof . Cantor also showed that sets with cardinality strictly greater than c {\displaystyle {\mathfrak {c}}} exist (see his generalized diagonal argument and theorem ). They include, for instance: Both have cardinality The cardinal equalities c 2 = c , {\displaystyle {\mathfrak {c}}^{2}={\mathfrak {c}},} c ℵ 0 = c , {\displaystyle {\mathfrak {c}}^{\aleph _{0}}={\mathfrak {c}},} and c c = 2 c {\displaystyle {\mathfrak {c}}^{\mathfrak {c}}=2^{\mathfrak {c}}} can be demonstrated using cardinal arithmetic : If A and B are disjoint sets , then From this, one can show that in general,
1980-415: Was not defined as a specific object itself. However, such an object can be defined as follows. The relation of having the same cardinality is called equinumerosity , and this is an equivalence relation on the class of all sets. The equivalence class of a set A under this relation, then, consists of all those sets which have the same cardinality as A . There are two ways to define the "cardinality of
2025-576: Was that the cardinality of the continuum ( c {\displaystyle {\mathfrak {c}}} ) is greater than that of the natural numbers ( ℵ 0 {\displaystyle \aleph _{0}} ); that is, there are more real numbers R than natural numbers N . Namely, Cantor showed that c = 2 ℵ 0 = ℶ 1 {\displaystyle {\mathfrak {c}}=2^{\aleph _{0}}=\beth _{1}} (see Beth one ) satisfies: The continuum hypothesis states that there
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