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68-508: (Redirected from Directs ) [REDACTED] Look up direct in Wiktionary, the free dictionary. Direct may refer to: Mathematics [ edit ] Directed set , in order theory Direct limit of (pre), sheaves Direct sum of modules , a construction in abstract algebra which combines several vector spaces Computing [ edit ] Direct access (disambiguation) ,

136-507: A {\displaystyle a} and b {\displaystyle b} are on opposite sides of x 0 . {\displaystyle x_{0}.} Explicitly, this happens when { a , b } = { x 0 − r , x 0 + r } {\displaystyle \{a,b\}=\left\{x_{0}-r,x_{0}+r\right\}} for some real r ≠ 0 , {\displaystyle r\neq 0,} in which case

204-519: A ≤ I b {\displaystyle a\leq _{I}b} and b ≤ I a {\displaystyle b\leq _{I}a} even though a ≠ b . {\displaystyle a\neq b.} Had this preorder been defined on R {\displaystyle \mathbb {R} } instead of R ∖ { x 0 } {\displaystyle \mathbb {R} \backslash \lbrace x_{0}\rbrace } then it would still form

272-577: A net is a function from a directed set and a sequence is a function from the natural numbers N . {\displaystyle \mathbb {N} .} Every sequence canonically becomes a net by endowing N {\displaystyle \mathbb {N} } with ≤ . {\displaystyle \,\leq .\,} If x ∙ = ( x i ) i ∈ I {\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}

340-409: A , b {\displaystyle a,b} and c {\displaystyle c} in X {\displaystyle X} : Reflexivity (1.) already follows from connectedness (4.), but is required explicitly by many authors nevertheless, to indicate the kinship to partial orders. Total orders are sometimes also called simple , connex , or full orders . A set equipped with

408-406: A Noetherian ring is a ring whose ideals satisfy the ascending chain condition. In other contexts, only chains that are finite sets are considered. In this case, one talks of a finite chain , often shortened as a chain . In this case, the length of a chain is the number of inequalities (or set inclusions) between consecutive elements of the chain; that is, the number minus one of elements in

476-401: A formal theory , which is a set of sentences with certain properties (details of which can be found in the article on the subject ). For instance, S {\displaystyle S} could be a first-order theory (like Zermelo–Fraenkel set theory ) or a simpler zeroth-order theory . The preordered set ( S , ⇐ ) {\displaystyle (S,\Leftarrow )}

544-1584: A generalized series of an I {\displaystyle I} -indexed collection of numbers ( r i ) i ∈ I {\displaystyle \left(r_{i}\right)_{i\in I}} (or more generally, the sum of elements in an abelian topological group , such as vectors in a topological vector space ) as the limit of the net of partial sums F ∈ Finite ⁡ ( I ) ↦ ∑ i ∈ F r i ; {\displaystyle F\in \operatorname {Finite} (I)\mapsto {\textstyle \sum \limits _{i\in F}}r_{i};} that is: ∑ i ∈ I r i   :=   lim F ∈ Finite ⁡ ( I )   ∑ i ∈ F r i   =   lim { ∑ i ∈ F r i : F ⊆ I , F  finite  } . {\displaystyle \sum _{i\in I}r_{i}~:=~\lim _{F\in \operatorname {Finite} (I)}\ \sum _{i\in F}r_{i}~=~\lim \left\{\sum _{i\in F}r_{i}\,:F\subseteq I,F{\text{ finite }}\right\}.} Let S {\displaystyle S} be

612-481: A chain can be identified with a monotone sequence , and is called an ascending chain or a descending chain , depending whether the sequence is increasing or decreasing. A partially ordered set has the descending chain condition if every descending chain eventually stabilizes. For example, an order is well founded if it has the descending chain condition. Similarly, the ascending chain condition means that every ascending chain eventually stabilizes. For example,

680-564: A corresponding total preorder on that subset. All definitions tacitly require the homogeneous relation R {\displaystyle R} be transitive : for all a , b , c , {\displaystyle a,b,c,} if a R b {\displaystyle aRb} and b R c {\displaystyle bRc} then a R c . {\displaystyle aRc.} A term's definition may require additional properties that are not listed in this table. A binary relation that

748-492: A directed set but it would now have a (unique) greatest element , specifically x 0 {\displaystyle x_{0}} ; however, it still wouldn't be partially ordered. This example can be generalized to a metric space ( X , d ) {\displaystyle (X,d)} by defining on X {\displaystyle X} or X ∖ { x 0 } {\displaystyle X\setminus \left\{x_{0}\right\}}

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816-497: A directed set by defining ( n 1 , n 2 ) ≤ ( m 1 , m 2 ) {\displaystyle \left(n_{1},n_{2}\right)\leq \left(m_{1},m_{2}\right)} if and only if n 1 ≤ m 1 {\displaystyle n_{1}\leq m_{1}} and n 2 ≤ m 2 . {\displaystyle n_{2}\leq m_{2}.} In analogy to

884-528: A directed set by writing U ≤ V {\displaystyle U\leq V} if and only if U {\displaystyle U} contains V . {\displaystyle V.} For every U , {\displaystyle U,} V , {\displaystyle V,} and W {\displaystyle W}  : The set Finite ⁡ ( I ) {\displaystyle \operatorname {Finite} (I)} of all finite subsets of

952-395: A generalization of convergent sequences. Totally ordered set In mathematics , a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation ≤ {\displaystyle \leq } on some set X {\displaystyle X} , which satisfies the following for all

1020-445: A given set that is ordered by inclusion, and the term is used for stating properties of the set of the chains. This high number of nested levels of sets explains the usefulness of the term. A common example of the use of chain for referring to totally ordered subsets is Zorn's lemma which asserts that, if every chain in a partially ordered set X has an upper bound in X , then X contains at least one maximal element. Zorn's lemma

1088-505: A greatest element is a directed set with the same preorder. For instance, in a poset P , {\displaystyle P,} every lower closure of an element; that is, every subset of the form { a ∈ P : a ≤ x } {\displaystyle \{a\in P:a\leq x\}} where x {\displaystyle x} is a fixed element from P , {\displaystyle P,}

1156-424: A least element. Thus every finite total order is in fact a well order . Either by direct proof or by observing that every well order is order isomorphic to an ordinal one may show that every finite total order is order isomorphic to an initial segment of the natural numbers ordered by <. In other words, a total order on a set with k elements induces a bijection with the first k natural numbers. Hence it

1224-501: A method of accessing data in a database Direct connect (disambiguation) , various methods of telecommunications and computer networking Direct memory access , access to memory by hardware subsystems independently of the CPU Entertainment [ edit ] Direct (Tower of Power album) Direct (Vangelis album) Direct (EP) , by The 77s Other uses [ edit ] Direct (music symbol) ,

1292-400: A music symbol used in music notation that is similar to a catchword in literature Nintendo Direct , an online presentation frequently held by Nintendo Mars Direct , a proposal for a crewed mission to Mars DIRECT , a proposed space shuttle-derived launch vehicle DirectX , a proprietary dynamic media platform Direct current , a direct flow of electricity Direct examination ,

1360-488: A partially ordered set that is not directed is the set { a , b } , {\displaystyle \{a,b\},} in which the only order relations are a ≤ a {\displaystyle a\leq a} and b ≤ b . {\displaystyle b\leq b.} A less trivial example is like the following example of the "reals directed towards x 0 {\displaystyle x_{0}} " but in which

1428-470: A set I {\displaystyle I} is directed with respect to ⊆ {\displaystyle \,\subseteq \,} since given any two A , B ∈ Finite ⁡ ( I ) , {\displaystyle A,B\in \operatorname {Finite} (I),} their union A ∪ B ∈ Finite ⁡ ( I ) {\displaystyle A\cup B\in \operatorname {Finite} (I)}

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1496-677: A set X {\displaystyle X} is a strict partial order on X {\displaystyle X} in which any two distinct elements are comparable. That is, a strict total order is a binary relation < {\displaystyle <} on some set X {\displaystyle X} , which satisfies the following for all a , b {\displaystyle a,b} and c {\displaystyle c} in X {\displaystyle X} : Asymmetry follows from transitivity and irreflexivity; moreover, irreflexivity follows from asymmetry. For delimitation purposes,

1564-425: A subset A {\displaystyle A} of a partially ordered set ( P , ≤ ) {\displaystyle (P,\leq )} is called a directed subset if it is a directed set according to the same partial order: in other words, it is not the empty set , and every pair of elements has an upper bound. Here the order relation on the elements of A {\displaystyle A}

1632-413: A total order as defined above is sometimes called non-strict order. For each (non-strict) total order ≤ {\displaystyle \leq } there is an associated relation < {\displaystyle <} , called the strict total order associated with ≤ {\displaystyle \leq } that can be defined in two equivalent ways: Conversely,

1700-434: A total order is a totally ordered set ; the terms simply ordered set , linearly ordered set , and loset are also used. The term chain is sometimes defined as a synonym of totally ordered set , but generally refers to a totally ordered subset of a given partially ordered set. An extension of a given partial order to a total order is called a linear extension of that partial order. A strict total order on

1768-486: Is a maximal element if for every j ∈ I , {\displaystyle j\in I,} m ≤ j {\displaystyle m\leq j} implies j ≤ m . {\displaystyle j\leq m.} It is a greatest element if for every j ∈ I , {\displaystyle j\in I,} j ≤ m . {\displaystyle j\leq m.} Any preordered set with

1836-640: Is a real number then the set I := R ∖ { x 0 } {\displaystyle I:=\mathbb {R} \backslash \lbrace x_{0}\rbrace } can be turned into a directed set by defining a ≤ I b {\displaystyle a\leq _{I}b} if | a − x 0 | ≥ | b − x 0 | {\displaystyle \left|a-x_{0}\right|\geq \left|b-x_{0}\right|} (so "greater" elements are closer to x 0 {\displaystyle x_{0}} ). We then say that

1904-688: Is a directed set because if A , B ∈ S {\displaystyle A,B\in S} and if C := A ∧ B {\displaystyle C:=A\wedge B} denotes the sentence formed by logical conjunction ∧ , {\displaystyle \,\wedge ,\,} then A ⇐ C {\displaystyle A\Leftarrow C} and B ⇐ C {\displaystyle B\Leftarrow C} where C ∈ S . {\displaystyle C\in S.} If S / ∼ {\displaystyle S/\sim }

1972-469: Is a directed set with respect to ⊇ ; {\displaystyle \,\supseteq ;\,} in fact, it is even a prefilter. If T {\displaystyle T} is a topological space and x 0 {\displaystyle x_{0}} is a point in T , {\displaystyle T,} the set of all neighbourhoods of x 0 {\displaystyle x_{0}} can be turned into

2040-426: Is a directed set with respect to the partial order ⊇ {\displaystyle \,\supseteq \,} and that also does not contain the empty set (this condition prevents triviality because otherwise, the empty set would then be a greatest element with respect to ⊇ {\displaystyle \,\supseteq \,} ). Every π -system , which is a non-empty family of sets that

2108-449: Is a directed set with respect to the partial order ⊇ {\displaystyle \,\supseteq \,} (respectively, ⊆ {\displaystyle \,\subseteq \,} ) if and only if the intersection (respectively, union) of any two of its members contains as a subset (respectively, is contained as a subset of) some third member. In symbols, a family I {\displaystyle I} of sets

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2176-505: Is a linear order, where the sets A i {\displaystyle A_{i}} are pairwise disjoint, then the natural total order on ⋃ i A i {\displaystyle \bigcup _{i}A_{i}} is defined by The first-order theory of total orders is decidable , i.e. there is an algorithm for deciding which first-order statements hold for all total orders. Using interpretability in S2S ,

2244-410: Is a natural order ≤ + {\displaystyle \leq _{+}} on the set A 1 ∪ A 2 {\displaystyle A_{1}\cup A_{2}} , which is called the sum of the two orders or sometimes just A 1 + A 2 {\displaystyle A_{1}+A_{2}} : Intuitively, this means that the elements of

2312-439: Is also possible to define a downward-directed set in which every pair of elements has a common lower bound. A subset of a poset is downward-directed if and only if its upper closure is a filter . Directed subsets are used in domain theory , which studies directed-complete partial orders . These are posets in which every upward-directed set is required to have a least upper bound . In this context, directed subsets again provide

2380-534: Is an upper bound of A {\displaystyle A} and B {\displaystyle B} in Finite ⁡ ( I ) . {\displaystyle \operatorname {Finite} (I).} This particular directed set is used to define the sum ∑ i ∈ I r i {\displaystyle {\textstyle \sum \limits _{i\in I}}r_{i}} of

2448-426: Is antisymmetric, transitive, and reflexive (but not necessarily total) is a partial order . A group with a compatible total order is a totally ordered group . There are only a few nontrivial structures that are (interdefinable as) reducts of a total order. Forgetting the orientation results in a betweenness relation . Forgetting the location of the ends results in a cyclic order . Forgetting both data results in

2516-512: Is any net from a directed set ( I , ≤ ) {\displaystyle (I,\leq )} then for any index i ∈ I , {\displaystyle i\in I,} the set x ≥ i := { x j : j ≥ i  with  j ∈ I } {\displaystyle x_{\geq i}:=\left\{x_{j}:j\geq i{\text{ with }}j\in I\right\}}

2584-457: Is called the tail of ( I , ≤ ) {\displaystyle (I,\leq )} starting at i . {\displaystyle i.} The family Tails ⁡ ( x ∙ ) := { x ≥ i : i ∈ I } {\displaystyle \operatorname {Tails} \left(x_{\bullet }\right):=\left\{x_{\geq i}:i\in I\right\}} of all tails

2652-550: Is closed under the intersection of any two of its members, is a directed set with respect to ⊇ . {\displaystyle \,\supseteq \,.} Every λ-system is a directed set with respect to ⊆ . {\displaystyle \,\subseteq \,.} Every filter , topology , and σ-algebra is a directed set with respect to both ⊇ {\displaystyle \,\supseteq \,} and ⊆ . {\displaystyle \,\subseteq \,.} By definition,

2720-465: Is common to index finite total orders or well orders with order type ω by natural numbers in a fashion which respects the ordering (either starting with zero or with one). Totally ordered sets form a full subcategory of the category of partially ordered sets , with the morphisms being maps which respect the orders, i.e. maps f such that if a ≤ b then f ( a ) ≤ f ( b ). A bijective map between two totally ordered sets that respects

2788-447: Is commonly used with X being a set of subsets; in this case, the upper bound is obtained by proving that the union of the elements of a chain in X is in X . This is the way that is generally used to prove that a vector space has Hamel bases and that a ring has maximal ideals . In some contexts, the chains that are considered are order isomorphic to the natural numbers with their usual order or its opposite order . In this case,

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2856-400: Is complete but the set of rational numbers Q is not. In other words, the various concepts of completeness (not to be confused with being "total") do not carry over to restrictions . For example, over the real numbers a property of the relation ≤ is that every non-empty subset S of R with an upper bound in R has a least upper bound (also called supremum) in R . However, for

2924-410: Is directed with respect to ⊇ {\displaystyle \,\supseteq \,} (respectively, ⊆ {\displaystyle \,\subseteq \,} ) if and only if or equivalently, Many important examples of directed sets can be defined using these partial orders. For example, by definition, a prefilter or filter base is a non-empty family of sets that

2992-512: Is directed. Every maximal element of a directed preordered set is a greatest element. Indeed, a directed preordered set is characterized by equality of the (possibly empty) sets of maximal and of greatest elements. The subset inclusion relation ⊆ , {\displaystyle \,\subseteq ,\,} along with its dual ⊇ , {\displaystyle \,\supseteq ,\,} define partial orders on any given family of sets . A non-empty family of sets

3060-543: Is given by regular chains of polynomials. Another example is the use of "chain" as a synonym for a walk in a graph . One may define a totally ordered set as a particular kind of lattice , namely one in which we have We then write a ≤ b if and only if a = a ∧ b {\displaystyle a=a\wedge b} . Hence a totally ordered set is a distributive lattice . A simple counting argument will verify that any non-empty finite totally ordered set (and hence any non-empty subset thereof) has

3128-417: Is inherited from P {\displaystyle P} ; for this reason, reflexivity and transitivity need not be required explicitly. A directed subset of a poset is not required to be downward closed ; a subset of a poset is directed if and only if its downward closure is an ideal . While the definition of a directed set is for an "upward-directed" set (every pair of elements has an upper bound), it

3196-814: Is sometimes called an upward directed set . A downward directed set is defined analogously, meaning that every pair of elements is bounded below. Some authors (and this article) assume that a directed set is directed upward, unless otherwise stated. Other authors call a set directed if and only if it is directed both upward and downward. Directed sets are a generalization of nonempty totally ordered sets . That is, all totally ordered sets are directed sets (contrast partially ordered sets , which need not be directed). Join-semilattices (which are partially ordered sets) are directed sets as well, but not conversely. Likewise, lattices are directed sets both upward and downward. In topology , directed sets are used to define nets , which generalize sequences and unite

3264-464: Is the Lindenbaum–Tarski algebra associated with S {\displaystyle S} then ( S / ∼ , ⇐ ) {\displaystyle \left(S/\sim ,\Leftarrow \right)} is a partially ordered set that is also a directed set. Directed set is a more general concept than (join) semilattice: every join semilattice is a directed set, as

3332-474: The monadic second-order theory of countable total orders is also decidable. There are several ways to take two totally ordered sets and extend to an order on the Cartesian product , though the resulting order may only be partial . Here are three of these possible orders, listed such that each order is stronger than the next: Each of these orders extends the next in the sense that if we have x ≤ y in

3400-845: The product order this is the product direction on the Cartesian product. For example, the set N × N {\displaystyle \mathbb {N} \times \mathbb {N} } of pairs of natural numbers can be made into a directed set by defining ( n 0 , n 1 ) ≤ ( m 0 , m 1 ) {\displaystyle \left(n_{0},n_{1}\right)\leq \left(m_{0},m_{1}\right)} if and only if n 0 ≤ m 0 {\displaystyle n_{0}\leq m_{0}} and n 1 ≤ m 1 . {\displaystyle n_{1}\leq m_{1}.} If x 0 {\displaystyle x_{0}}

3468-401: The reflexive closure of a strict total order < {\displaystyle <} is a (non-strict) total order. The term chain is sometimes defined as a synonym for a totally ordered set, but it is generally used for referring to a subset of a partially ordered set that is totally ordered for the induced order. Typically, the partially ordered set is a set of subsets of

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3536-450: The unit interval [0,1], and the affinely extended real number system (extended real number line). There are order-preserving homeomorphisms between these examples. For any two disjoint total orders ( A 1 , ≤ 1 ) {\displaystyle (A_{1},\leq _{1})} and ( A 2 , ≤ 2 ) {\displaystyle (A_{2},\leq _{2})} , there

3604-594: The chain. Thus a singleton set is a chain of length zero, and an ordered pair is a chain of length one. The dimension of a space is often defined or characterized as the maximal length of chains of subspaces. For example, the dimension of a vector space is the maximal length of chains of linear subspaces , and the Krull dimension of a commutative ring is the maximal length of chains of prime ideals . "Chain" may also be used for some totally ordered subsets of structures that are not partially ordered sets. An example

3672-862: The existence of an upper bound of the empty subset implies that A {\displaystyle A} is nonempty. The set of natural numbers N {\displaystyle \mathbb {N} } with the ordinary order ≤ {\displaystyle \,\leq \,} is one of the most important examples of a directed set. Every totally ordered set is a directed set, including ( N , ≤ ) , {\displaystyle (\mathbb {N} ,\leq ),} ( N , ≥ ) , {\displaystyle (\mathbb {N} ,\geq ),} ( R , ≤ ) , {\displaystyle (\mathbb {R} ,\leq ),} and ( R , ≥ ) . {\displaystyle (\mathbb {R} ,\geq ).} A (trivial) example of

3740-429: The in-trial questioning of a witness by the party who has called him or her to testify See also [ edit ] All pages with titles beginning with Direct All pages with titles containing Direct Direction (disambiguation) Director (disambiguation) Indirect (disambiguation) Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with

3808-419: The join or least upper bound of two elements is the desired c . {\displaystyle c.} The converse does not hold however, witness the directed set {1000,0001,1101,1011,1111} ordered bitwise (e.g. 1000 ≤ 1011 {\displaystyle 1000\leq 1011} holds, but 0001 ≤ 1000 {\displaystyle 0001\leq 1000} does not, since in

3876-404: The last bit 1 > 0), where {1000,0001} has three upper bounds but no least upper bound, cf. picture. (Also note that without 1111, the set is not directed.) The order relation in a directed set is not required to be antisymmetric , and therefore directed sets are not always partial orders . However, the term directed set is also used frequently in the context of posets. In this setting,

3944-419: The order topology on N induced by < and the order topology on N induced by > (in this case they happen to be identical but will not in general). The order topology induced by a total order may be shown to be hereditarily normal . A totally ordered set is said to be complete if every nonempty subset that has an upper bound , has a least upper bound . For example, the set of real numbers R

4012-487: The ordering rule only applies to pairs of elements on the same side of x 0 {\displaystyle x_{0}} (that is, if one takes an element a {\displaystyle a} to the left of x 0 , {\displaystyle x_{0},} and b {\displaystyle b} to its right, then a {\displaystyle a} and b {\displaystyle b} are not comparable, and

4080-458: The preorder a ≤ b {\displaystyle a\leq b} if and only if d ( a , x 0 ) ≥ d ( b , x 0 ) . {\displaystyle d\left(a,x_{0}\right)\geq d\left(b,x_{0}\right).} An element m {\displaystyle m} of a preordered set ( I , ≤ ) {\displaystyle (I,\leq )}

4148-403: The product order, this relation also holds in the lexicographic order, and so on. All three can similarly be defined for the Cartesian product of more than two sets. Applied to the vector space R , each of these make it an ordered vector space . See also examples of partially ordered sets . A real function of n real variables defined on a subset of R defines a strict weak order and

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4216-412: The rational numbers this supremum is not necessarily rational, so the same property does not hold on the restriction of the relation ≤ to the rational numbers. There are a number of results relating properties of the order topology to the completeness of X: A totally ordered set (with its order topology) which is a complete lattice is compact . Examples are the closed intervals of real numbers, e.g.

4284-453: The reals have been directed towards x 0 . {\displaystyle x_{0}.} This is an example of a directed set that is neither partially ordered nor totally ordered . This is because antisymmetry breaks down for every pair a {\displaystyle a} and b {\displaystyle b} equidistant from x 0 , {\displaystyle x_{0},} where

4352-451: The second set are added on top of the elements of the first set. More generally, if ( I , ≤ ) {\displaystyle (I,\leq )} is a totally ordered index set, and for each i ∈ I {\displaystyle i\in I} the structure ( A i , ≤ i ) {\displaystyle (A_{i},\leq _{i})}

4420-473: The subset { a , b } {\displaystyle \{a,b\}} has no upper bound). Let D 1 {\displaystyle \mathbb {D} _{1}} and D 2 {\displaystyle \mathbb {D} _{2}} be directed sets. Then the Cartesian product set D 1 × D 2 {\displaystyle \mathbb {D} _{1}\times \mathbb {D} _{2}} can be made into

4488-468: The title Direct . 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=Direct&oldid=1243204563 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Directed set The notion defined above

4556-446: The two orders is an isomorphism in this category. For any totally ordered set X we can define the open intervals We can use these open intervals to define a topology on any ordered set, the order topology . When more than one order is being used on a set one talks about the order topology induced by a particular order. For instance if N is the natural numbers, < is less than and > greater than we might refer to

4624-443: The various notions of limit used in analysis . Directed sets also give rise to direct limits in abstract algebra and (more generally) category theory . In addition to the definition above, there is an equivalent definition. A directed set is a set A {\displaystyle A} with a preorder such that every finite subset of A {\displaystyle A} has an upper bound. In this definition,

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