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order, in taxonomy: see classification.

The Columbia Electronic Encyclopedia Copyright © 2004.

Licensed from Columbia University Press

Licensed from Columbia University Press

or **rules of order**

Generally accepted rules, precedents, and practices used in the governance of deliberative assemblies. They are intended to maintain decorum, ascertain the will of the majority, preserve the rights of the minority, and facilitate the orderly transaction of business. Rules of parliamentary procedure originated in Britain in the 16th and 17th centuries and were subsequently adopted by legislatures around the world. *Robert's Rules of Order*, codified in 1876 by U.S. Gen. Henry M. Robert (1837–1923) and regularly refined and enlarged, is the standard set of rules used by legislatures in the U.S.

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Certificate requiring the issuer to pay a certain sum of money on demand to a specific person or organization. Money orders provide a fast, safe, and convenient means of transferring small sums of money. They are issued by governments (usually through postal authorities), banks, and other qualified institutions to buyers who pay the issuer the face amount of the money order plus a service charge. Because they are exchangeable for cash on demand, they are a generally accepted means of payment. The American Express Co. began issuing money orders in 1882; the company also created the first traveler's checks nine years later. *Seealso* currency.

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In Britain, a regulation traditionally issued by the sovereign on the advice of the Privy Council. In modern practice, an order is issued only on the advice of ministers, and the minister in charge of the department concerned with the order's subject matter is responsible to Parliament for its contents. Most orders in council today are issued to implement legislation passed by Parliament.

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Capital styles for the five major orders of Classical architecture.

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or **giant order**Architectural order in which the columns extend beyond one interior story, often through several stories. Though giant columns were used in antiquity, they were first applied to building facades in Renaissance Italy. Any of the orders might be treated in this manner. The colossal order was revived in 18th-century Europe.

Colossal order, court facade of Blenheim Palace, Oxfordshire, England, by Sir John Vanbrugh, begun *elipsis*

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In civil proceedings, a court order compelling a party to do or to refrain from doing a specified act. It is an equitable remedy for harm for which no adequate remedy exists in law. Thus it is used to prevent a future harmful action (e.g., disclosing confidential information, instituting a national labour strike, or violating a group's civil rights) rather than to compensate for an injury that has already occurred. It also provides relief from harm for which an award of money damages is not a satisfactory solution. A defendant who violates an injunction may be cited for contempt. *Seealso* equity.

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or **Teutonic Knights** officially **House of the Hospitallers of Saint Mary of the Teutons**

Religious order important in eastern Europe in the late Middle Ages. Founded in 1189–90 to nurse the sick in Palestine during the Third Crusade, it was militarized in 1198 and given land in Jerusalem and Germany. It transferred its base of operations to eastern Europe in the 13th century, gaining control of Prussia by 1283 and making Marienburg the centre of a military principality (1309–1525). The order extended its influence until it was defeated at the Battle of Tannenberg (1410). Another Polish victory in 1466 forced the knights to cede lands to Poland and become vassals of the Polish king. In 1525 the grand master in Prussia converted to Protestantism, dissolved the order in Prussia, and became a duke under Polish suzerainty. In other parts of Europe, especially Austria, the order survived the Reformation. Napoleon declared the order dissolved in 1809 and redistributed most of its remaining lands. In 1834 the Austrian emperor refounded it as a charitable religious order, and it is now headquartered in Vienna.

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Member of the Order of Friars Preachers, a Roman Catholic preaching and teaching order founded by St. Dominic. It dates officially from 1216, when Pope Honorius III gave it his approval, though Dominic had begun to build it at least a decade earlier. In contrast with earlier orders, the Dominicans were not organized in autonomous monastic houses, but joined the order at large and could be sent to any house or province on the order's business. The early teaching centers were at Bologna, Paris, Cologne, and Oxford. Dominican preachers spoke out against the Cathari, the Moors, and the Jews and were among the first missionaries under the Spanish and Portuguese explorers. Dominicans were put in charge of the Inquisition when it was founded. Perhaps their most famous member was St. Thomas Aquinas.

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In mathematics, a well-order relation (or well-ordering) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering.
Equivalently, a well-ordering is a well-founded total order.
The set S together with the well-order relation is then called a well-ordered set.## Ordinal numbers

## Examples

## Properties

## Equivalent formulations

## Order topology

Every well-ordered set can be made into a topological space by endowing it with the order topology.## See also

Every element, except a possible greatest element, has a unique successor (next element). Every subset which has an upper bound has a least upper bound. There may be elements (besides the least element) which have no predecessor.

Spelling note: The hyphen is frequently omitted in contemporary papers, yielding the spellings wellorder, wellordered, and wellordering.

Every well-ordered set is uniquely order isomorphic to a unique ordinal number, called the order type of the well-ordered set. The position of each element within the ordered set is also given by an ordinal number. In the case of a finite set, the basic operation of counting, to find the ordinal number of a particular object, or to find the object with a particular ordinal number, corresponds to assigning ordinal numbers one by one to the objects. The size (number of elements, cardinal number) of a finite set is equal to the order type. Counting in the everyday sense typically starts from one, so it assigns to each object the size of the initial segment with that object as last element. Note that these numbers are one more than the formal ordinal numbers according to the order isomorphy, because these are equal to the number of earlier objects (which corresponds to counting from zero). Thus for finite n, the expression "n-th element" of a well-ordered set requires context to know whether this counts from zero or one. In a notation "β-th element" where β can also be an infinite ordinal, it will typically count from zero.

For an infinite set the order type determines the cardinality, but not conversely: well-ordered sets of a particular cardinality can have many different order types. For a countably infinite set the set of possible order types is even uncountable.

- The standard ordering ≤ of the natural numbers is a well-ordering.
- The standard ordering ≤ of the integers is not a well-ordering, since, for example, the set of negative integers does not contain a least element.
- The following relation R is an example of well-ordering of the integers: x R y if and only if one of the following conditions holds:
- x = 0
- x is positive, and y is negative
- x and y are both positive, and x ≤ y
- x and y are both negative, and y ≤ x
- :R can be visualized as follows:
- ::0 1 2 3 4 ..... -1 -2 -3 .....
- :R is isomorphic to the ordinal number ω + ω.
- Another relation for well-ordering the integers is the following definition: x <
_{z}y iff |x| < |y| or (|x| = |y| and x ≤ y). This well-order can be visualized as follows:

- 0 -1 1 -2 2 -3 3 -4 4 ...

- The standard ordering ≤ of the positive real numbers is not a well-ordering, since, for example, the open interval (0, 1) does not contain a least element. From the ZFC axioms of set theory (including the axiom of choice) one can show that there is a well-order of the reals; it is also possible to show that the ZFC axioms alone are not sufficient to prove the existence of a definable (by a formula) well-order of the reals. However it is consistent with ZFC that a definable well-ordering of the reals exists—for example, it is consistent with ZFC that V=L, and it follows from ZFC+V=L that a particular formula well-orders the reals, or indeed any set.
- An uncountable subset of real numbers with the standard "≤" is not well-ordered because the real line can only contain countably many disjoint intervals. A countably infinite subset may or may not be a well-order with the standard "≤". Examples of well-orders:
- The set of numbers { - 2
^{-n}| 0 ≤ n < ω } has order type ω. - The set of numbers { - 2
^{-n}- 2^{-m-n}| 0 ≤ m,n < ω } has order type ω². The previous set is the set of limit points within the set. Within the set of real numbers, either with the ordinary topology or the order topology, 0 is also a limit point of the set. It is also a limit point of the set of limit points. - The set of numbers { - 2
^{-n}| 0 ≤ n < ω } ∪ { 1 } has order type ω + 1. With the order topology of this set, 1 is a limit point of the set. With the ordinary topology (or equivalently, the order topology) of the real numbers it is not.

In a well-ordered set, every element, unless it is the overall largest, has a unique successor: the smallest element that is larger than it. However, not every element needs to have a predecessor. As an example, consider an ordering of the natural numbers where all even numbers are less than all odd numbers, and the usual ordering applies within the evens and the odds:

- 0 2 4 6 8 ... 1 3 5 7 9 ...

This is a well-ordered set of order type ω + ω. Note that while every element has a successor (there is no largest element), two elements lack a predecessor: 0 and 1.

If a set is well-ordered, the proof technique of transfinite induction can be used to prove that a given statement is true for all elements of the set.

The well-ordering theorem, which is equivalent to the axiom of choice, states that every set can be well-ordered. The well-ordering theorem is also equivalent to the Kuratowski-Zorn lemma.

In a well-ordered set, every subset with an upper bound has a supremum.

If a set is totally ordered, then the following are equivalent:

- The set is well-ordered. That is, every nonempty subset has a least element.
- Transfinite induction works for the entire ordered set.
- Every strictly decreasing sequence of elements of the set must terminate after only finitely many steps (assuming the axiom of dependent choice).

With respect to this topology there can be two kinds of elements:

- isolated points - these are the minimum and the elements with a predecessor
- limit points - this type does not occur in finite sets, and may or may not occur in an infinite set; the infinite sets without limit point are the sets of order type ω, for example N.

For subsets we can distinguish:

- Subsets with a maximum (that is, subsets which are by itself); this can be an isolated point or a limit point of the whole set; in the latter case it may or may not be also a limit point of the subset.
- Subsets which are unbounded by itself but bounded in the whole set; they have no maximum, but a supremum outside the subset; if the subset is non-empty this supremum is a limit point of the subset and hence also of the whole set; if the subset is empty this supremum is the minimum of the whole set.
- Subsets which are unbounded in the whole set.

A subset is cofinal in the whole set if and only if it is unbounded in the whole set or it has a maximum which is also maximum of the whole set.

A well-ordered set as topological space is a first-countable space if and only if it has order type less than or equal to ω_{1} that is, if and only if the set is countable or has the smallest uncountable order type.

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Last updated on Friday September 19, 2008 at 14:26:38 PDT (GMT -0700)

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