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Free

Specialty Definition: Free

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Specialty Definition: Free

(From Wikipedia, the free Encyclopedia)

See:

• freedom, libre
• free software
• Free (band)
• gratis, free as in beer
• free object as in mathematics, with
  • free group
  • free abelian group
  • free module

being important special cases.

Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Free."

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Free (band)

(From Wikipedia, the free Encyclopedia)

Free was a R&B-styled rock band which formed in London in 1968.

Members

• Paul Rodgers -- Lead Vocals
• Paul Kossoff -- Lead Guitar
• Andy Fraser -- Bass Guitar
• Simon Kirke -- Drums

Discography

• Tons of Sobs (1968)
• Free (1969)
• Fire and Water (1970)
• Highway (1970)
• Free Live! (1971) (live)
• Kossoff, Kirke, Tetsu & Rabbit (1971) [1]
• Free at Last (1972)
• Heartbreaker (1973)

[1] Although having members of Free, it is not regarded as an official Free album.

The following albums were issued after the band ceased recording:

• The Free Story (1974)
• The Best of Free (1975)
• Free and Easy, Rough and Ready (1976)
• Completely Free (1982)
• All Right Now: The Best of Free (1991)
• Molten Gold: The Anthology (1994) (2 disc set)
• Songs of Yesterday (2000) (5 disc box set)

see also: Bad Company

Stub alert!

Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Free (band)."

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Free as in beer

(From Wikipedia, the free Encyclopedia)

The expression free as in beer refers to things which are available at no monetary cost (like free beer at a party). It can be contrasted with the expressions free as in speech or free as in freedom, which refer to something which is free of restrictions, as in the freedom of speech.

Since the advent of the free software movement, these terms have entered frequent use, for categorising computer programs according to the licenses and legal fetters which cover them. The expression "free as in freedom" is also the title of a 2002 biography of Richard Stallman, founder of the Free Software Foundation.

In French (and other Latin and shemic languages), the distinction is simpler, because the word free can be translated as gratuit (no cost, gratis in Latin) or libre (free of restrictions).

To some people (outside the particular context of software), "free beer" can imply free riding, or freeloading taking advantage of something that is not paid for.

Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Free as in beer."

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Free group

(From Wikipedia, the free Encyclopedia)

In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many elements of S and their inverses (disregarding trivial variations such as st^-1 = su^-1ut^-1). Note that the notion of free group is different from the notion free abelian group: in this case the order in the product matters.

Examples

The group (Z,+) of integers is free; we can take S = {1}. A free group on a two-element subset S occurs in the proof of the Banach-Tarski paradox and is described there.

Construction

If S is any set, there always exists a free group on S. This free group on S is essentially unique in the following sense: if F₁ and F₂ are two free groups on the set S, then F₁ and F₂ are isomorphic, and furthermore there exists precisely one group isomorphism f : F₁ -> F₂ such that f(s) = s for all s in S.

This free group on S is denoted by F(S) and can be constructed as follows. For every s in S, we introduce a new symbol s^-1. We then form the set of all finite strings consisting of symbols of S and their inverses. Two such strings are considered equivalent if one arises from the other by replacing two adjacent symbols ss^-1 or s^-1s by the empty string. This generates an equivalence relation on the set of strings; its quotient set is defined to be F(S). Because the equivalence relation is compatible with string concatenation, F(S) becomes a group with string concatenation as operation.

If S is the empty set, then F(S) is the trivial group consisting only of its identity element.

Universal property

The free group on S is characterized by the following universal property: if G is any group and f : S -> G is any function, then there exists a unique group homomorphism T : F(S) -> G such that T(s) = f(s) for all s in S.

Free groups are thus instances of the more general concept of free objects in category theory. Like all universal constructions, they give rise to a pair of adjoint functors.

Facts and theorems

Any group G is a quotient group of some free group F(S). If S can be chosen to be finite here, then G is called finitely generated.

Any subgroup of a free group is free (Nielsen-Schreier theorem).

Any connected graph can be viewed as a path-connected topological space by treating an edge between two vertices as a continuous path between those vertices. With this understanding, the fundamental group of every connected graph is free. This fact can be used to prove the Nielsen-Schreier theorem.

If F is a free group on S and also on T, then S and T have the same cardinality. This cardinality is called the rank of the free group F.

If S has more than one element, then F(S) is not abelian, and in fact the center of F(S) is trivial (that is, consists only of the identity element).

Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Free group."

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Free market

(From Wikipedia, the free Encyclopedia)

A free-market (free-trade or neo-liberal) economy is an idealized form of a market economy in which buyers and sellers are permitted to carry out transactions based on mutual agreement on price without government intervention in the form of taxes, subsidies, regulation, or government ownership of goods or services. The free market is considered the mainstay of ideologies such as minarchism and libertarianism and Western definitions of capitalism. It is anathema to communism and some variants of socialism, as defined in the West, although most variants of socialism seek to mitigate what they see as the problems of an unrestrained free market.

In reality there are no totally free or ideal markets in operation. Lack of perfect knowledge, monopolistic practices, cartels, taxes and government regulation bias the equilibrium points of most large markets in existence today. Participants engage in information bias practices such as insider trading and price fixing. Some believe that the notion of a free market is inherently inachievable because they believe that governments are fundamentally involved in markets through the creation and enforcement of property rights. Others argue that the concept of property comes from natural law and therefore it is incorrect to see governments as creating markets.

In the ideal free market, the law of supply and demand functions, influencing prices toward an equilibrium that balances the demands for the products against the supplies. At these equilibrium prices, the market distributes the products to the purchasers according to each purchaser's use (or utility) for each product and within the relative limits of each buyer's purchasing power. In the limited mathematical ideal market this distribution of products is Pareto Optimal (see Pareto efficiency), meaning that no purchaser could have their purchasing limits filled in a way more useful to them without reducing the usefulness of some other purchaser's bundle of products. This type of optimality doesn't necessarily have anything to say about the distribution of purchasing power itself (which is often an input to the mathematical ideal market) - the optimality generally refers to the distribution of products given the pre-existing purchasing power of the purchasers. The necessary components for the functioning of such a free market include no artificial price pressures from taxes, subsidies, tariffs, or government regulation, perfect (or at a minimum, equivalent) knowledge about the value of the goods, geographic availablity of goods to all people, and no artificial monopolies or other forms of economic coercion on the part of the actors.

The distribution of purchasing power in an economy depends to a large extent on the Labor market and the Financial markets, but also on other things such as family relationships, inheritance, gifts and so on. The ideal free market does not explain particularly well the performance of many real markets such as the Labor market, or Financial markets. It explains better the markets for consumers products.

The economic and political application of the concept of the ideal free market is known, primarily by detractors, as neoliberalism.

See: Austrian School, Friedrich Hayek, Adam Smith, command economy, capitalism, socialism, communism

See also: Free software, Nash equilibrium, Game theory

External links

Definitions and Distinctions

Free Market

Capitalism

Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Free market."

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Free module

(From Wikipedia, the free Encyclopedia)

A free module is a module having a free basis.

For an R-module M, the set E = {e₁, e₂, ... e_n} is a free basis for M if and only if:

1) E is a generating set for M, that is to say every element of M is a sum of elements of E multiplied by coefficients in R.

2) if r₁e₁ + r₂e₂ + ... + r_ne_n = 0, then r₁ = r₂ = ... = r_n = 0 (where 0 is the identity element of M and 0 is the identity element of R).

If M has a free basis with n elements, then M is said to be free of rank n, or more generally free of finite rank.

Note that an immediate corollary of (2) is that the coefficients in (1) are unique for each x.

The definition of an infinite free basis is similar, except that E will have infinitely many elements. In general, the summation which generates the elements x of M may be infinite but must converge in whatever sense is appropriate for M. For some modules this will mean that it must be a finite sum, and thus that for any particular x only finitely many of the elements of E are involved.

In the case of an infinite basis, the rank of M is the cardinality of E.

Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Free module."

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Free object

(From Wikipedia, the free Encyclopedia)

The idea of a free object in mathematics is one of the basics of abstract algebra. It is part of universal algebra, in the sense that it relates to all types of algebraic structure (with finitary operations); but on the other hand it has a clean formulation in terms of category theory (in yet more abstract terms). It is probably better to master some special case such as free groups first.

Starting from a familiar concept in group theory, of defining a group 'by generators and relations', we can say that in general a free object of a certain specific algebraic type will have 'generators and no relations'. Yet. If we want to use the method of generators and relations in generality, we split the approach up as (i) create an object with the generators which are left as general as possible; and then (ii) impose relations, in the form of an equivalence relation which is a congruence. Therefore in these terms from universal algebra we need to understand free objects for step (i), and the nature of congruences for step (ii).

An example would be free monoids. These are rather simpler than free groups: the free monoid on a set X, is the monoid of all finite strings using X as alphabet, with operation concatenation of strings. The identity is the empty string. (See Kleene star.)

As that example suggests, free objects look like constructions from syntax; and we can reverse that to some extent by saying that major uses of syntax can be explained and characterised as free objects, in a way that makes apparently heavy 'punctuation' explicable (and more memorable). An example for that is the way a free magma on X turns out to be the magma of binary trees labelled at the leaves by X. That construction generalises (from a single binary operation to any collection of 'arities') in a way the free object concept makes much more palatable. (See also Herbrand universe.)

In general, the setting for a free object is like this: a category C of algebraic structures (sets plus operations, obeying some laws) has a functor F to Sets, the category of sets and functions, that simply ignores the operations. We call F a forgetful functor. Free objects are created by a left adjoint G to F: for a set X the free object on X as 'generators' is G(X). There are general existence theorems that apply.

Other types of forgetfulness also give rise to objects quite like free objects: for example the tensor algebra construction on a vector space as left adjoint to the function on algebras over a field that ignores the algebra structure.

Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Free object."

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Free software

(From Wikipedia, the free Encyclopedia)

The term free software is used in essentially two different ways:

1. any software which may be copied and used without payment (think free beer)
2. software that can be copied, used, studied, modified, distributed, etc., with few or no restrictions (think free speech and free market).

These definitions may conflict and a piece of software that is free in the first sense may not be free in the second, and vice versa.

Free software of the "free speech" type is sometimes called "software libre", from the French "logiciel libre" and the Spanish "software libre". In fact, in many languages there isn't this conflict between free as in freedom and free as in "free beer": "libre" translates to "free" in the sense of "freedom". Free software of the other type is called "gratis", which translates to the "free" of "free beer".

"Free Beer" definition

Various types of free software in this sense exists:

• freeware, software that can be distributed and used without cost. Few strings are attached; sometimes only private, non-commercial use is allowed. The software may not be modified, and sometimes may also not be redistributed.
• adware, freeware which displays advertisements during use. It is often installed without the consent of the installee.
• spyware, collects market research data and/or credit card numbers from the host computer. Also often (if not always) installed without consent.
• crippleware, software which can be used in a limited form for free; the enhanced version typically requires payment (see shareware).
• abandonware, software which is used and distributed in violation of copyright license, but for which copyright is not enforced any more

Shareware is not a type of free software, since its license requires payment for use beyond a specified trial period. The payment typically has to be made by the user on an "honor system". Warez, software which is distributed for free by a third party in violation of its copyright license, is also not considered to be free software in this sense.

"Free Speech" definition

The freedom definition of "free software" has been championed by the Free Software Foundation (FSF) of notable software author Richard Stallman, who codified his philosophy of software freedom in the 1980s.

The FSF has produced a specific free software definition; a software is "free" in this sense if it grants:

1. the freedom to run the program for any purpose
2. the freedom to study and modify the program
3. the freedom to copy the program
4. the freedom to redistribute modified or unmodified versions of the program

Number 2 requires access to the program's source code.

A list of compliant licenses is available from FSF's web site (see below). The term "proprietary software" is used for software distributed under more restrictive licenses which don't grant these freedoms. Usually, copyright law reserves most rights of modification, duplication and redistribution for the copyright owner; software released under a free software license specifically rescinds most of these reserved rights.

The FSF definition of free software does not touch on the issue of price; a commonly used slogan is "free as in speech, not as in beer", and it is common to see CDs of free software such as Linux distributions for sale. However, in this situation the buyer of the CD would have the right to copy and distribute his CD. The FSF definition of free software in fact can conflict with the free beer definition of software. Many free beer software applications forbid the end user from commercially profiting from the software or otherwise charging for the software. This conflicts with the right to redistribute.

To avoid confusion, some people use the words "libre" and "gratis" to avoid the ambiguity of the English word "free". However, these alternative terms are still used mostly within the free software movement and are only slowly spreading to the outside world. Others advocate the term open source software but the relationship between this term of free software is complex and will be more fully explained below.

There are many variations on free software:

• Public domain software, in which the author has abandoned the copyright. Public-domain software, since it is not protected by copyright at all, may be freely incorporated into closed, proprietary works as well as free ones.
• BSD-style licensess, so called because they are applied to much of the software distributed with the BSD operating systems. The author under such licenses retains copyright protection solely to disclaim warranty and to require proper attribution of modified works, but permits redistribution and modification, even in proprietary works.
• Copyleft licenses, the most prominent of which are the GNU General Public License and the GNU Lesser General Public License. The author retains copyright, and permits redistribution and modification under terms designed to ensure that all modified versions of the software remain under copyleft terms.

See free software licenses for more information.

Note that the original copyright owner of copyleft-licensed software can also make a modified version under their original copyright, and sell it under any license they like, in addition to distributing the original version as free software. This technique has been used as a business model by a number of free software companies; this does not restrict any of the rights granted to the users of the copyleft version.

A large, and ever-growing, amount of software is made available under free software licenses; observers of this trend (and adherents to it) often refer to this phenomenon as the free software movement. Notable free software projects include the Linux and BSD operating system kernels, the GCC compiler, GDB debugger and C libraries, the BIND name server, the Sendmail mail transport server, the Apache web server, the MySQL and PostgreSQL relational database systems, the Perl, Python, Tcl and PHP programming languages, the X Window System, the GNOME and KDE desktop environments, the OpenOffice office suite, the Mozilla web browser and the GIMP graphics editor.

Like all free software, these projects distribute their programs under licenses that grant users all the freedoms discussed above, but because of technicalities in the licenses, combining programs by mixing source code or directly linking binaries may be problematic unless both applications are under mutually compatible licenses.

However, when programs are not directly linked together into a single program, these problems do not exist. Much free software can run on non-free platforms such as Microsoft Windows, and non-free software can be run on free platforms, although purists prefer to use all-free software running on a free platform such as Linux. Free software packages constitute a software ecosystem where different pieces of software can provide services to one another, leading to co-evolution of features: in one simple example, the Python programming language provides support for the HTTP protocol, and the Apache web server that provides the HTTP protocol can call the Python programming language to serve dynamic content.

The Debian Project, which produces an operating system completely composed of free software, created a set of guidelines called which are used to evaluate the freedom of a license before including software licensed like that into Debian. The Debian Free Software Guidelines cover:

• free redistribution
• inclusion of source code
• modifications and derived works
• integrity of the author's source code (as a compromise for the likes of TeX)
• no discrimination against persons or groups
• no discrimination against fields of endeavor, like commercial use
• distribution of license, it needs to apply to all to whom the program is redistributed
• license must not be specific to Debian, basically a reiteration of the last point
• license must not contaminate other software
• example licenses -- GPL, BSD, Artistic

Debian has collected over seven and a half thousand software packages compliant with the above guidelines (2003).

Recently, Debian developers have started arguing for the application of the same principles not only in software, but in software documentation as well. Many documents written by the Linux Documentation Project and some documents licensed under the GNU Free Documentation License do not comply with all of the above guidelines.

Comparison with Open Source software

The Open Source movement, which is philosophically distinct from the free software movement, was created by a group of people who formed the Open Source Initiative (OSI). They sought (1) to bring a higher profile to the practical benefits of sharing software source code, and (2) to interest major software houses and other high-tech industry companies in the concept. These advocates see the term open source as avoiding the ambiguity of the English word "free" in free software. Many people recognise a qualitative benefit to the software development process when a program's source code can be used, modified and redistributed by developers.

Since the OSI places emphasis on the pragmatic benefits of access to the program's source code, rather than focusing on user and programmer freedoms, the FSF has distanced itself both from the Open Source movement and from the term "Open Source". The free software movement places primary emphasis on the moral or ethical aspects of software, seeing technical excellence as a desirable by-product of its ethical standard. The Open Source movement sees technical excellence as the primary goal, regarding source code sharing as a means to an end.

In all cases, licenses which qualify as free software licenses also qualify as open source licenses. However, the reverse is a different matter since the Open Source Definition (OSD) does not explicitly and unambiguously state a requirement that open source licenses grant people the right to copy their software. For example, nowhere in the OSD or its rationale is the word "copy" included. Rather, some interpret the OSD as treating software like cars which you can inspect, tinker, modify and even resell ("redistribute"), while making copies is a different matter which the OSD never addresses. Note, however, that many interpret the term "redistribution" as used in the OSD to include copying. (The OSD is a modified form of the Debian Free Software Guidelines.)

If the OSD is treated as a distribution scheme, as Richard Stallman holds, then the right to copy software is necessarily implied by any OSD license. This view is strengthened by statements made by backers of the OSD that the term "open source software" is simply a "marketing campaign for Free Software". However, the proliferation of use licenses (notably by Microsoft) has led many people to believe that a license is required to run software. From that perspective, the OSD (by itself) does not grant nor imply any right to copy software unless the term "redistribute" is interpreted as including the act of copying.

Since the OSI only approves free software licenses as complying with the OSD, most people interpret it as a distribution scheme, and freely interchange 'open source' with 'free software'. And even though there are important philosophical differences between the two terms, particularly in terms of the motivations for developing and using such software, they seldom make any impact in the collaboration process.

While the term "Open Source" removes the ambiguity of Freedom versus Price, it introduces another - between programs that meet the Open Source Definition, giving users the freedom to improve upon them, and programs that simply have source available, possibly with heavy restrictions on the use of that source. Many people believe that any software that has source available is open source, because they can tinker with it themselves. However, much of this software does not give its users the freedom to distribute their modifications, restricts commercial usage, or otherwise removes users' rights.

Political significance

Since free software allows free use, modification, and distribution, it often finds a home in third world countries for whom the cost of proprietary software is sometimes too high. It is also easily modified locally, so translation efforts into languages which are not necessarily commercially profitable are also feasible. See also internationalization.

Most free software is produced by international teams cooperating through free association. Teams typically are composed of individuals with a wide variety of motivations. There are many stances about the relation of free software to the current, capitalist economic system:

• Some consider free software to be a competitor to capitalism
• Some consider free software to be another form of competition within free markets.
• Groups like Oekonux consider that everything could be produced in this manner and that this mode of production has the potential to superceed the capitalist mode of production.

See also

• Free audio software
• List of free game software

External links and references

• The FSF's free software definition
• FSF's philosophy collection on software and information freedoms
• FSF list of free software licenses
• The Open Source Initiative
• Why Open Source Software / Free Software (OSS/FS)? Look at the Numbers! - David Wheeler's quantitative analysis of the advantages of open-source software by an exhaustive review of published studies, analyses, and news stories.

  Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Free software."

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Free software license

(From Wikipedia, the free Encyclopedia)

Generally speaking, free software license is a phrase used by the free software movement to mean any software license that grants users of the software the following four freedoms:

1. The freedom to run the program for any purpose
2. The freedom to study and modify the program
3. The freedom to copy the program
4. The freedom to redistribute modified versions of the program

A license which preserves those freedoms for modified works is a copyleft license. See Free software movement for more information.

The Free Software Foundation maintains a list of free software licenses at their web site. The list distinguishes between free software licenses that are compatible or incompatible with the FSF license of choice, the GNU General Public License, which is a copyleft license. The list also contains licenses which the FSF considers non-free for various reasons. The list, which differs slightly from the open source license list, can be found at http://www.gnu.org/philosophy/license-list.html

See also:

• Free software
• Free Software Foundation
• Open source license

  Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Free software license."

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Free trade

(From Wikipedia, the free Encyclopedia)

Free trade is an economic concept referring to the selling of products between countries without tariffs or other trade barriers. International trade is often constricted by different national taxes, other fees imposed on exported and imported goods, as well as non-tariff regulations on imported goods; free trade is against all these restrictions.

Some multi-national entities, such as the European Union, have implemented free trade in some forms between their member nations (customs union). However, there is continuing debate whether free trade would help third world nations with different economic problems and whether free trade is good for the developed world.

Arguments for and against free trade

Many economists argue that free trade increases the standard of living through the law of comparative advantage and economies of scale. Others argue that free trade allows developed nations to exploit developing nations and to destroy local industry in addition to circumventing social and labor standards. Conversely it has also been argued that free trade hurts developed nations because it causes jobs from those nations to move to other countries as well as producing a race to the bottom which causes a general lowering of health and safety standards.

In addition, the current implementation of free trade has been criticized by advocates of free trade itself. One complaint is that developed nations tend to insist that developing nations open their markets to industrial products from the developed world, yet refuse to open their markets to agricultural goods from the developing world. Furthermore it has been noted that the current concept of free trade supports the free movement of products and employers, which favors the developed nations, but not the free movement of employees (i.e., labor), which would favor the people of developing nations. (See also: Immigration.)

Intellectual property and free trade

Historically, the free trade movement was skeptical and even hostile to the notion of intellectual property, regarded as monopolistic and harmful to a free, competitive economy. Indeed, during the late 19th century, free trade advocates succeeded in reducing the length of the patents available in many European countries. The Netherlands even abolished its patent system (temporarily, as it turned out).

The 19th century anti-patent cause failed largely because the recession of 1874 discredited the free trade movement of the time (and also because patent advocates used a public relations campaign which was remarkably sophisticated for its time).

It is thus quite remarkable (some would even say ironic) that corporations lobbying for expanded intellectual property privileges have succeeded in including TRIPS, a very strong treaty on intellectual property rights, as a membership requirement for the World Trade Organization, the international organization dedicated to furthering the cause of free trade.

See also

• Free market
• International trade
• Privatization

References

1. Fritz Machlup & Edith Penrose, "The Patent Controversy in the 19th Century", Journal of Economic History, 10 (1) pp 1-29, 1950.

Links

Pro-free trade/free-market

• Arguments for Free Trade from the Mises Institute
• Lewrockwell.com
• The Cato Institute's Free Trade website
• Techcentralstation.com
• The Heritage Foundation
• The Future of Freedom Foundation
• The Heartland Institute
• The Foundation for Economic Education
• The American Enterprise Institute
• The Timbro Institute (Sweden)

Opponents of free trade

• United States Reform Party
• Ross Perot
• Pat Buchanan
• AFL-CIO
• Ralph Nader
• US Green Party
• Indymedia.org

  Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Free trade."

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Freedom

(From Wikipedia, the free Encyclopedia)

Freedom has various uses:

• Freedom the political right, or legal capacity, of self-determination, as an expression of the individual will
• Freedom of the will as a human capacity considered philosophically
• Freedom of religion
• Freedom of speech
• Freedom of the press
• Freedom in the context of software with "no restrictions on use" for any purpose
• Freedom, a British Anarchist newspaper
• Freedom a town in New York
• Freedom, a US Space station incorporated into the International Space Station in November 1993

See also: free, free software, liberty, Human rights

Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Freedom."

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Group action

(From Wikipedia, the free Encyclopedia)

In mathematics, groups are often used to describe symmetries of objects. This is formalized by the notion of a group action: every element of the group "acts" like a bijective map (or "symmetry") on some set. In this case, the group is also called a transformation group of the set.

Definition

If G is a group and X is a set, then a (left) group action of G on X is a binary function G × X -> X (where the image of g in G and x in X is written as g.x) which satisfies the following two axioms:

1. g.(h.x) = (gh).x for all g, h in G and x in X.
2. e.x = x for every x in X; here e denotes the identity element of G.

From these two axioms, it follows that for every g in G, the function which maps x in X to g.x is a bijective map from X to X. Therefore, one may alternatively and equivalently define a group action of G on X as a group homomorphism G -> Sym(X), where Sym(X) denotes the group of all bijective maps from X to X.

If a group action G × X -> X is given, we also say that G acts on the set X or X is a G-set.

In complete analogy, one can define a right group action of G on X as a function X × G -> X by the two axioms (x.g).h = x.(gh) and x.e = x. In the sequel, we consider only left group actions.

Examples

• Every group G acts on G in two natural ways: g.x = (gx) for all x in G, or g.x = (gxg ^-1) for all x in G.
• The symmetric group S_n and its subgroups act on the set { 1, ... , n } by permutating its elements.
• The symmetry group of a polyhedron acts on the set of vertices of that polyhedron.
• The symmetry group of any geometrical object acts on the set of points of that object.
• The automorphism group of a vector space (or graph, or group, or ring...) acts on the vector space (or set of vertices of the graph, or group, or ring...).
• The Lie groups Gl(n,R), SL(n,R) and O(n,R) act on Rⁿ.
• The Galois group of a field extension E/F acts on the bigger field E. So does every subgroup of the Galois group.
• The additive group of the real numbers (R, +) acts on the phase space of "well-behaved" systems in classical mechanics (and in more general dynamical systems): if t is in R and x is in the phase space, then x describes a state of the system, and t.x is defined to be the state of the system t seconds later if t is positive or -t seconds ago if t is negative.

Types of actions

The action of G on X is called

• transitive if for any two x, y in X there exists an g in G such that g.x = y;
• simply transitive if for any two x, y in X there exists precisely one g in G such that g.x = y.
• faithful (or effective) if for any two different g, h in G there exists an x in X such that g.x ≠ h.x;
• free if for any two different g, h in G and all x in X we have g.x ≠ h.x;

Every free action on a non-empty set is faithful. A group G that acts faithfully on a set X is isomorphic to a permutation group on X. An action is regular if and only if it is transitive and free.

Orbits and stabilizers

If we define N = {g in G : g.x = x for all x in X}, then N is a normal subgroup of G and the factor group G/N acts faithfully on X by setting (gN).x = g.x. The action of G on X is faithful if and only if N = {e}.

If Y is a subset of X, we write GY for the set { g.y : y in Y and g in G}. We call the subset Y invariant under G if GY = Y (which is equivalent to GY ⊆ Y). In that case, G also operates on Y. The subset Y is called fixed under G if g.y = y for all g in G and all y in Y. Every subset that's fixed under G is also invariant under G, but not vice versa.

Any operation of G on X defines an equivalence relation on X: two elements x and y are called equivalent if there exists a g in G with g.x = y. The equivalence class of x under this equivalence relation is given by the set Gx = { g.x : g in G } which is also called the orbit of x. The elements x and y are equivalent if and only if their orbits are the same: Gx = Gy. Every orbit is an invariant subset of X on which G acts transitively. The action of G on X is transitive if and only if all elements are equivalent, meaning that there is only one orbit. The set of all orbits is written as X/G.

For every x in X, we define G_x = { g in G : g.x = x }. This is a subgroup of G, and it is called the stabilizer of x or isotropy subgroup at x. The action of G on X is free if and only if all stabilizers consist only of the identity element.

There is a natural bijection between the set of all left cosets of the subgroup G_x and the orbit of x, given by hG_x |-> h.x. Therefore, |Gx| = [G : G_x], and so

This result, known as the orbit-stabilizer theorem, is especially useful if G and X are finite, because then it can be employed for counting arguments. A related result is Burnside's lemma:

where r is the number of orbits, and X^g is the set of points fixed by g. This result too is mainly of use when G and X are finite, when it can be interpreted as follows: the number of orbits is equal to the average number of points fixed per group element.

Morphisms and isomorphisms between G-sets

If X and Y are two G-sets, we define a morphism from X to Y to be a function f : X -> Y such that f(g.x) = g.f(x) for all g in G and all x in X. If such a function f is bijective, then its inverse is also a morphism, and we call f an isomorphism and the two G-sets X and Y are called isomorphic; for all practical purposes, they are indistinguishable in this case.

Some example isomorphisms:

• Every regular G action is isomorphic to the action of G on G given by left multiplication.
• Every free G action is isomorphic to G×S, where S is some set and G acts by left multiplication on the first coordinate.
• Every transitive G action is isomorphic to left multiplication by G on the set of left cosets of some subgroup H of G.

With this notion of morphism, the collection of all G-sets forms a category; this category is a topos.

Generalizations

One often considers continuous group actions: the group G is a topological group, X is a topological space, and the map G × X → X is continuous with respect to the product topology of G × X. The space X is also called a G-space in this case. This is indeed a generalization, since every group can be considered a topological group by using the discrete topology. All the concepts introduced above still work in this context, however we define morphisms between G-spaces to be continuous maps compatible with the action of G. The above statements about isomorphisms for regular, free and transitive actions are no longer valid for continuous group actions.

One can also consider actions of monoids on sets, by using the same two axioms as above. This does not define bijective maps and equivalence relations however.

Instead of actions on sets, one can define actions of groups and monoids on objects of an arbitrary category: start with an object X of some category, and then define an action on X as a monoid homomorphism into the monoid of endomorphisms of X. If X has an underlying set, then all definitions and facts stated above can be carried over. For example, if we take the category of vector spaces, we obtain group representations in this fashion.

One can view a group G as a category with a single object in which every morphism is invertible. A group action is then nothing but a functor from G to the category of sets, and a group representation is a functor from G to the category of vector spaces. In analogy, an action of a groupoid is a functor from the groupoid to the category of sets or to some other category.

Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Group action."

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